Depth-one foliations, pseudo-Anosov flows and universal circles

Junzhi Huang

doi:10.4171/ggd/964

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Depth-one foliations, pseudo-Anosov flows and universal circles

Junzhi Huang
Yale University, New Haven, USA

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Abstract

Given a taut depth-one foliation

ℱ

in a closed atoroidal 3-manifold transverse to a pseudo-Anosov flow

ϕ

without perfect fits, we show that the universal circle coming from leftmost sections associated to

ℱ,

constructed by Thurston and Calegari–Dunfield, is isomorphic to Fenley’s ideal boundary of the flow space associated to

ϕ

with natural structure maps. As a corollary, we use a theorem of Barthelmé–Frankel–Mann to show that there is at most one pseudo-Anosov flow without perfect fits transverse to

ℱ

up to orbit equivalence.

1. Introduction

There has been an important theme in 3-manifold topology to study the interaction between flows and codimension-one foliations in 3-manifolds. The simplest examples of codimension-one foliations of a 3-manifold

M

are fibrations, which are exactly the foliations with all leaves compact. The theory of Thurston norm organizes different ways of fibration of

M

into a finite number of fibered faces, and there is a one-to-one correspondence between fibered faces and isotopy classes of suspension pseudo-Anosov flows [22

D. Fried, Fibrations over

S^{1}

with pseudo-Anosov monodromy. In Travaux de Thurston sur les surfaces, pp. 51–266, Astérisque 66–67, Société Mathématique de France, Paris, 1979 Zbl 0446.57023 MR 0568308

, 31

W. P. Thurston, A norm for the homology of

3 -

manifolds. Mem. Amer. Math. Soc. 59 (1986), no. 339, 99–130 Zbl 0585.57006 MR 0823443

]. The aim of this paper is to study one of the next simplest classes of foliations, namely depth-one foliations, and their interaction with transverse pseudo-Anosov flows, by comparing the

π_{1} -

actions on

S^{1}

that arise in both settings.

A foliation in a closed 3-manifold is called a depth-one foliation if the restriction to the complement of compact leaves is a fibration over the circle. More precisely, there are a finite number of compact leaves, called depth-zero leaves, and the rest of the leaves (namely the depth-one leaves) are infinite-type surfaces spiraling into the depth-zero leaves. One way to construct depth-one foliations is to “spin” a fibration around an embedded surface (see [6

D. Calegari, Foliations and the geometry of 3-manifolds. Oxford Math. Monogr., Oxford University Press, Oxford, 2007, 363 pp. Zbl 1118.57002 MR 2327361

, Example 4.8]).

Given a taut depth-one foliation

ℱ

in a closed 3-manifold

M,

a result of Candel [8

A. Candel, Uniformization of surface laminations. Ann. Sci. Éc. Norm. Super. (4) 26 (1993), no. 4, 489–516 Zbl 0785.57009 MR 1235439

] shows that there exists a Riemannian metric on

M

such that the restrictions to the leaves of

ℱ

are hyperbolic, giving every

ℱ -

leaf a standard hyperbolic structure in the sense of [10

J. Cantwell and L. Conlon, Hyperbolic geometry and homotopic homeomorphisms of surfaces. Geom. Dedicata 177 (2015), no. 1, 27–42 Zbl 1359.37097 MR 3370020

] (see also Section 2.3). In particular, there is a natural circle at infinity associated to the universal circle of any leaf of

ℱ .

An unpublished construction of Thurston [32

W. P. Thurston, Three-manifolds, foliations and circles, II. Unfinished manuscript, 1998

], which was later written down by Calegari–Dunfield [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

], produces a circle

𝔖_{left}

associated to

ℱ .

We will call this circle a universal circle from leftmost sections. The circle

𝔖_{left}

is acted on faithfully by

π_{1} (M)

and is equipped with a

π_{1} (M) -

equivariant collection of monotone structure maps

{U_{λ}}_{λ \in \tilde{ℱ}}

to the circles at infinity of all

\tilde{ℱ} -

leaves, where

\tilde{ℱ}

is the lift of

ℱ

to the universal cover

\tilde{M}

of

M .

In general, there is an axiomatized notion of a universal circle associated to a taut foliation (Definition 4.1). The universal circle

𝔖_{left}

is a universal circle of

ℱ

in this general sense but not a canonical one. However, when

ℱ

is a taut depth-one foliation transverse to a pseudo-Anosov flow without perfect fits

ϕ

(see Section 2.1 for discussions on pseudo-Anosov flows), we will see that it is possible to relate the

𝔖_{left}

to a more natural object, which is the ideal boundary of the flow space of

ϕ .

For a pseudo-Anosov flow

ϕ,

the flow space

𝒪

associated to

ϕ

is the space of orbits of the lifted flow

\tilde{ϕ}

in

\tilde{M} .

It is homeomorphic to

ℝ^{2}

by [2

T. Barbot, Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles. Ergodic Theory Dynam. Systems 15 (1995), no. 2, 247–270 Zbl 0826.58025 MR 1332403

, 14

S. Fenley and L. Mosher, Quasigeodesic flows in hyperbolic 3-manifolds. Topology 40 (2001), no. 3, 503–537 Zbl 0990.53040 MR 1838993

, 16

S. R. Fenley, Anosov flows in

3 -

manifolds. Ann. of Math. (2) 139 (1994), no. 1, 79–115 Zbl 0796.58039 MR 1259365

], and there is a compactification

\bar{𝒪} = 𝒪 \cup \partial 𝒪

given by Fenley [13

S. Fenley, Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry. Geom. Topol. 16 (2012), no. 1, 1–110 Zbl 1279.37026 MR 2872578

]. The ideal boundary

\partial 𝒪

is homeomorphic to a circle, and the

π_{1} (M) -

action on

𝒪

extends continuously to

\partial 𝒪 .

If

ϕ

has no perfect fits, we will see that the shadow of any leaf

λ

of

\tilde{ℱ}

provides a natural structure map

I_{λ}

from

\partial 𝒪

to the circle at infinity of

λ

(see Section 3). For these structure maps, we prove the following theorem.

Theorem 1.1.

Let $M$ be a closed atoroidal $3 -$ manifold with a pseudo-Anosov flow $ϕ$ without perfect fits, and let $ℱ$ be a taut depth-one foliation in $M$ transverse to $ϕ .$ Then the circle $\partial 𝒪,$ together with the structure maps ${I_{λ}}_{λ \in \tilde{ℱ}},$ is a universal circle for $ℱ .$

While writing this paper, the author learned that Landry, Minsky and Taylor show a much more general version of Theorem 1.1 (recently appeared in [24

M. P. Landry, Y. N. Minsky, and S. J. Taylor, Simultaneous universal circles. J. Topol. 19 (2026), no. 1, article no. e70054, 28 pp. Zbl 08143171 MR 5009301

]). More precisely, they prove that given a taut foliation

ℱ

almost transverse to a pseudo-Anosov flow

ϕ

in a closed hyperbolic manifold

M,

the boundary of the flow space naturally has the structure of a universal circle for

ℱ .

Nevertheless, we show in our setting that the universal circles

\partial 𝒪

and

𝔖_{left}

are isomorphic, as made precise by the following theorem.

Theorem 1.2.

Let $M$ be a closed atoroidal $3 -$ manifold with a pseudo-Anosov flow $ϕ$ without perfect fits, and let $ℱ$ be a taut depth-one foliation in $M$ transverse to $ϕ .$ Then the $π_{1} (M) -$ actions on $\partial 𝒪$ and on $𝔖_{left}$ are conjugated by a homeomorphism $T : 𝔖_{left} \to \partial 𝒪 .$ Moreover, for any leaf $λ$ of $\tilde{ℱ},$ we have $I_{λ} \circ T = U_{λ} .$

Corollary 1.3.

Let $M$ be a closed atoroidal $3 -$ manifold, and let $ℱ$ be a taut depth-one foliation in $M .$ Then there is at most one pseudo-Anosov flow without perfect fits transverse to $ℱ$ up to orbit equivalence.

Proof.

Suppose there are two pseudo-Anosov flows without perfect fits

ϕ

and

φ

that are transverse to

ℱ .

Since

M

is atoroidal, both

ϕ

and

φ

are transitive (we say a flow is transitive if it has an orbit that is dense in both positive and negative time) [27

L. Mosher, Dynamical systems and the homology norm of a

3 -

manifold, I: efficient intersection of surfaces and flows. Duke Math. J. 65 (1992), no. 3, 449–500 Zbl 0754.58030 MR 1154179

]. Since the construction of

𝔖_{left}

makes no use of the transverse flows, the actions of

π_{1} (M)

on the ideal boundaries of the orbit spaces of

ϕ

and

φ

are conjugate by Theorem 1.2. By [3

T. Barthelmé, S. Frankel, and K. Mann, Orbit equivalences of pseudo-Anosov flows. Invent. Math. 240 (2025), no. 3, 1119–1192 Zbl 08038253 MR 4902161

, Theorem 1.5],

ϕ

and

φ

are orbit equivalent.

Remark 1.4.

In personal conversations, Michael Landry told the author that using the construction by Gabai–Mosher for almost transverse pseudo-Anosov flows to finite depth foliations, one can construct different pseudo-Anosov flows transverse to the same depth-one foliation, but these flows have perfect fits. While there is currently no complete proof of Gabai–Mosher’s construction in the literature, the monograph [28

L. Mosher, Laminations and flows transverse to finite depth foliations. Preprint, 1996

] by Mosher contains an outline and the main ideas of the theory. See also the paper [25

M. P. Landry and C. C. Tsang, Endperiodic maps, splitting sequences, and branched surfaces. Geom. Topol. 29 (2025), no. 9, 4531–4663 Zbl 08152702 MR 5017754

] by Landry–Tsang and their upcoming work [26

M. P. Landry and C. C. Tsang, Pseudo-Anosov flows from sutured hierarchies. In preparation

], which are aimed at revisiting the theory using veering triangulations.

Remark 1.5.

In [30

A. Parlak, Mutations and faces of the Thurston norm ball dynamically represented by multiple distinct flows. Geom. Topol. 29 (2025), no. 4, 2105–2173 Zbl 08084075 MR 4929474

], Anna Parlak constructs examples of closed and cusped hyperbolic 3-manifolds with a non-fibered face dynamically represented by two topologically inequivalent pseudo-Anosov flows using mutations of veering triangulations. However, the properties of the resulting flows, for example, whether they have perfect fits or which foliations they are transverse to, are not so clear.

A conjectural picture of “pseudo-Anosov packages” is developed in [5

D. Calegari, Promoting essential laminations. Invent. Math. 166 (2006), no. 3, 583–643 Zbl 1106.57014 MR 2257392

] by Calegari in the hope that the different structures from taut foliations, laminations, universal circles and pseudo-Anosov flows are organized and compatible in the most natural way, and it is asked to what extent the picture is true.

In particular, given a universal circle

𝔖

for

ℱ,

he constructs a pair of

π_{1} (M) -

invariant laminations

Ξ^{\pm}

on

𝔖 .

In our case, one can apply the construction to the universal circle

𝔖_{left} ≅ \partial 𝒪

and get a pair of laminations on

\partial 𝒪 .

On the other hand, the endpoints of the singular foliations

ℱ_{𝒪}^{u / s}

also induce a pair of laminations

ℒ_{𝒪}^{u / s}

on

\partial 𝒪

by taking the pairs of endpoints of regular leaves and faces of singular leaves. We partially verify Calegari’s picture by showing that

Ξ^{\pm}

and

ℒ_{𝒪}^{u / s}

coincide.

Theorem 1.6.

In the setting of Theorem 1.2, the invariant lamination $Ξ^{+}$ (resp. $Ξ^{-}$ ) on the universal circle $𝔖_{left}$ equals the induced stable lamination $ℒ_{𝒪}^{s}$ (resp. $ℒ_{𝒪}^{u}$ ) under the isomorphism $T : 𝔖_{left} \to \partial 𝒪 .$

The organization of the paper is as follows. In Section 2, we briefly recall some knowledge about pseudo-Anosov flows, depth-one foliations and circle laminations. In Section 3, we summarize the structure of the shadows of leaves of

ℱ

in

𝒪

developed by [12

D. Cooper, D. D. Long, and A. W. Reid, Bundles and finite foliations. Invent. Math. 118 (1994), no. 1, 255–283 Zbl 0858.57015 MR 1292113

, 19

S. R. Fenley, Surfaces transverse to pseudo-Anosov flows and virtual fibers in

3 -

manifolds. Topology 38 (1999), no. 4, 823–859 Zbl 0926.57009 MR 1679801

] and carefully study the infinity of shadows. From there, we introduce the restriction maps

I_{λ}

and a relative version of restriction maps. We start Section 4 with a brief review of the construction of the universal circle from leftmost sections

𝔖_{left}

following [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

], and we relate

𝔖_{left}

to the universal circle structure of

\partial 𝒪

we developed in Section 3. We prove Theorem 1.2 in Section 5 by explicitly constructing the homeomorphism

T

and proving the desired properties. We conclude with a discussion of invariant laminations and the proof of Theorem 1.6 in Section 6.

2. Preliminaries

Convention 2.1.

We consider a closed Riemannian atoroidal

3 -

manifold

M

with the Riemannian metric to be determined later. For a partition,

Θ

(e.g., a flow or a foliation) of a space

X

and a point

x \in X,

we let

Θ (x)

be the atom of

Θ

containing

x .

More generally, if

A

is a subset of

C,

we use

Θ (A)

to denote the saturation of

A

by

Θ -

atoms.

2.1. Pseudo-Anosov flows

We refer to [1

I. Agol and C. C. Tsang, Dynamics of veering triangulations: infinitesimal components of their flow graphs and applications. Algebr. Geom. Topol. 24 (2024), no. 6, 3401–3453 Zbl 1552.57031 MR 4812222

, 14

S. Fenley and L. Mosher, Quasigeodesic flows in hyperbolic 3-manifolds. Topology 40 (2001), no. 3, 503–537 Zbl 0990.53040 MR 1838993

, 27

L. Mosher, Dynamical systems and the homology norm of a

3 -

manifold, I: efficient intersection of surfaces and flows. Duke Math. J. 65 (1992), no. 3, 449–500 Zbl 0754.58030 MR 1154179

] and the recent monograph [4

T. Barthelmé and K. Mann, Pseudo-Anosov flows: a plane approach. [v1] 2025 [v2] 2026, arXiv:2509.15375v2

] for detailed discussions of pseudo-Anosov flows in 3-manifolds. The following definition follows [13

S. Fenley, Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry. Geom. Topol. 16 (2012), no. 1, 1–110 Zbl 1279.37026 MR 2872578

].

A flow

ϕ : M \times ℝ \to M

in

M

is a pseudo-Anosov flow if it has the following properties:

each flowline is $C^{1}$ and not a single point;
the tangent line bundle $T ϕ$ is continuous;
there are a finite number of singular closed orbits, and a pair of 2-dimensional singular foliations $ℱ^{u}$ and $ℱ^{s}$ in $M$ so that
- each leaf of $ℱ^{u}$ or $ℱ^{s}$ is a union of $ϕ -$ orbits;
- outside of the singular orbits $ℱ^{u}$ and $ℱ^{s}$ are regular foliations whose leaves intersect transversely along $ϕ -$ orbits;
- for each singular orbit $ω,$ the leaf of $ℱ^{u}$ or $ℱ^{s}$ containing $ω$ is homeomorphic to $P_{n} \times [0, 1] / f$ where
  
  $P_{n} = {r e^{2 k i π / n} ∣ r \geq 0, 0 \leq k \leq n - 1} \subset ℂ$
  
  is an $n -$ prong and $f$ is a homeomorphism from $P_{n}$ to $P_{n} .$ The orbit $ω$ is the image of ${0} \times [0, 1]$ and $n$ is always greater than 2 in our case;
- orbits in the same $ℱ^{s} -$ leaf are forward asymptotic, and orbits in the same $ℱ^{u} -$ leaf are backward asymptotic.

The singular foliations

ℱ^{s}

and

ℱ^{u}

are called the stable foliation and the unstable foliation of

ϕ,

respectively. When the set of singular orbits is empty,

ϕ

is an Anosov flow.

Fix a universal cover

\tilde{M}

of

M,

and let

\tilde{ϕ}, {\tilde{ℱ}}^{u}, {\tilde{ℱ}}^{s}

be the lifts of

ϕ, ℱ^{u}, ℱ^{s}

to

\tilde{M},

respectively. The quotient of

\tilde{M}

by

\tilde{ϕ}

is the flow space

𝒪,

which is homeomorphic to

ℝ^{2}

[2

T. Barbot, Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles. Ergodic Theory Dynam. Systems 15 (1995), no. 2, 247–270 Zbl 0826.58025 MR 1332403

, 14

S. Fenley and L. Mosher, Quasigeodesic flows in hyperbolic 3-manifolds. Topology 40 (2001), no. 3, 503–537 Zbl 0990.53040 MR 1838993

, 16

S. R. Fenley, Anosov flows in

3 -

manifolds. Ann. of Math. (2) 139 (1994), no. 1, 79–115 Zbl 0796.58039 MR 1259365

]. We orient

𝒪

so that the coorientation coincides with the flow direction, and the pictures in this paper are drawn in a way that the flow is flowing toward the reader. The deck transformation on

\tilde{M}

descends to an orientation-preserving

π_{1} (M) -

action on

𝒪,

and the singular foliations

{\tilde{ℱ}}^{u}

and

{\tilde{ℱ}}^{s}

descend to a pair of

π_{1} (M) -

invariant transverse singular foliations on

𝒪,

denoted by

ℱ_{𝒪}^{u}

and

ℱ_{𝒪}^{s} .

The singular leaves of

ℱ_{𝒪}^{s}

and

ℱ_{𝒪}^{u}

are

n -

pronged with

n \geq 3 .

The union of two adjacent prongs in a singular leaf of

ℱ_{𝒪}^{s}

or

ℱ_{𝒪}^{u}

is called a face.

A

\tilde{ϕ} -

orbit

ω

is periodic if there exists a non-trivial deck transformation

g

of

\tilde{M}

with

g (ω) = ω .

In this case,

g

acts on

ω

by translation, and

ω

covers a closed orbit of

ϕ

in

M .

Similarly, a leaf of

{\tilde{ℱ}}^{s}

or

{\tilde{ℱ}}^{u}

is called periodic if it is fixed by some non-trivial deck transformation of

\tilde{M} .

In particular, singular leaves are periodic. Finally, we call a point or a leaf of

ℱ_{𝒪}^{u}

or

ℱ_{𝒪}^{s}

in

𝒪

periodic if the corresponding

\tilde{ϕ} -

orbit or

{\tilde{ℱ}}^{s / u}

leaf in

\tilde{M}

is periodic. The following fact is well known and can be found in [4

T. Barthelmé and K. Mann, Pseudo-Anosov flows: a plane approach. [v1] 2025 [v2] 2026, arXiv:2509.15375v2

, Proposition 1.4.3].

Lemma 2.2.

If $ℓ$ is a leaf of $ℱ_{𝒪}^{s}$ or $ℱ_{𝒪}^{u}$ fixed by a non-trivial $g$ in $π_{1} (M),$ then the $g -$ action on $ℓ$ has a unique fixed point. In other words, any periodic leaf of ${\tilde{ℱ}}^{s}$ or ${\tilde{ℱ}}^{u}$ contains a unique periodic orbit.

A ray of

ℱ_{𝒪}^{u}

or

ℱ_{𝒪}^{s}

is an embedded closed half-line contained in a leaf with the interior disjoint from singularities. Two rays

l \in ℱ_{𝒪}^{s}

and

l^{'} \in ℱ_{𝒪}^{u}

are said to form a perfect fit if there is an (possibly orientation-reversing) embedding

ι : [0, 1] \times [0, 1] - (1, 1) \to 𝒪

mapping horizontal lines to

ℱ_{𝒪}^{s}

leaves, vertical lines to

ℱ_{𝒪}^{u}

leaves,

[0, 1) \times {1}

to

l^{'}

and

{1} \times [0, 1)

to

l .

We say a pseudo-Anosov flow is without perfect fits if no two rays in

ℱ_{𝒪}^{s}

and

ℱ_{𝒪}^{u}

form a perfect fit. The notion of perfect fits is introduced and studied by Fenley in [17

S. R. Fenley, The structure of branching in Anosov flows of

3 -

manifolds. Comment. Math. Helv. 73 (1998), no. 2, 259–297 Zbl 0999.37008 MR 1611703

, 18

S. R. Fenley, Foliations with good geometry. J. Amer. Math. Soc. 12 (1999), no. 3, 619–676 Zbl 0930.53024 MR 1674739

]. In particular, we have the following lemma which is an immediate consequence of [18

S. R. Fenley, Foliations with good geometry. J. Amer. Math. Soc. 12 (1999), no. 3, 619–676 Zbl 0930.53024 MR 1674739

, Theorem 4.8].

Lemma 2.3.

If $ϕ$ has no perfect fits, then any non-trivial element of $π_{1} (M)$ has at most one fixed point in $𝒪 .$

Fenley introduces a compactification of

𝒪

in [13

S. Fenley, Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry. Geom. Topol. 16 (2012), no. 1, 1–110 Zbl 1279.37026 MR 2872578

] by building an ideal boundary

\partial 𝒪

homeomorphic to

S^{1},

and the resulting compactified space

\bar{𝒪} = 𝒪 \cup \partial 𝒪

is homeomorphic to a closed 2-disk. We orient

\partial 𝒪

as the boundary of

𝒪,

and the action of

π_{1} (M)

on

𝒪

extends continuously to an orientation-preserving action on

\bar{𝒪} .

Each ray in

ℱ_{𝒪}^{s}

or

ℱ_{𝒪}^{u}

has a well-defined endpoint, and the endpoints of every leaf are distinct. When the flow has no perfect fits, a ray in

ℱ_{𝒪}^{s}

and a ray in

ℱ_{𝒪}^{u}

always have distinct endpoints. If we moreover assume that

ϕ

is not conjugate to an Anosov suspension flow (which is automatic when

M

is atoroidal), the action of

π_{1} (M)

on

\partial 𝒪

is minimal [13

S. Fenley, Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry. Geom. Topol. 16 (2012), no. 1, 1–110 Zbl 1279.37026 MR 2872578

, Main Theorem].

Convention 2.4.

We assume that

ϕ

is a pseudo-Anosov flow without perfect fits in

M .

2.2. End-periodic automorphisms

We briefly recall the basics of end-periodic automorphisms of infinite-type surfaces, which arise naturally in the study of depth-one foliations. Readers are referred to [11

J. Cantwell, L. Conlon, and S. R. Fenley, Endperiodic automorphisms of surfaces and foliations. Ergodic Theory Dynam. Systems 41 (2021), no. 1, 66–212 Zbl 1464.57022 MR 4190052

] for a more complete treatment of the theory.

Let

L

be an infinite-type surface without boundary with finitely many ends, all of which are non-planar. Given a homeomorphism

f : L \to L,

an end

E

of

L

is a contracting end of

f

if there is a neighborhood

U_{E}

of

E

and an integer

n > 0

such that

f^{n} (U_{E}) ⊊ U_{E}

and

⋂_{k \geq 0} f^{n k} (U_{E})

is empty. Such a neighborhood

U_{E}

is called a regular neighborhood of

E .

An end is a repelling end of

f

if it is a contracting end of

f^{- 1},

and a regular neighborhood of

E

for

f

is just a regular neighborhood for

f^{- 1} .

A homeomorphism

f

is called end periodic if each end of

L

is either contracting or repelling. If

f

is end periodic, a multi-curve

δ

in

L

is called an $f -$ juncture if

δ

is the boundary of a regular neighborhood of an end. If the end is contracting,

δ

is called a positive

f -

juncture. Otherwise, it is a negative

f -

juncture. An

f -

invariant choice of a positive (resp. negative)

f -

juncture for each contracting (resp. repelling) end is called a system of positive (resp. negative) $f -$ junctures. An end-periodic homeomorphism is atoroidal if it does not preserve any essential multi-curve up to isotopy.

Given an end-periodic homeomorphism

f,

we fix a regular neighborhood

U_{E}

for every end

E .

Let

U^{+}

be the union of

U_{E}

of contracting ends, and let

U^{-}

be the union of those of repelling ends. The positive escaping set

𝒰^{+}

and the negative escaping set

𝒰^{-}

are defined as

𝒰^{\pm} = ⋃_{n \geq 0} f^{\mp n} (U^{\pm}) .

In other words,

𝒰^{+}

is the set of points whose positive iterations escape to contracting ends, and

𝒰^{-}

is the set of points whose negative iterations escape to repelling ends. The mapping torus

M_{f}

is non-compact, but it is topologically tame and possesses a nice compactification which we describe below.

The mapping torus

M_{f}

is the quotient of

L \times ℝ

by an automorphism

F

where

F

is given by

\begin{matrix} F : L \times ℝ & \to L \times ℝ \\ (x, t) & \mapsto (f^{- 1} (x), t + 1) . \end{matrix}

We attach

𝒰^{+} \times {+ \infty}

and

𝒰^{-} \times {- \infty}

to

L \times ℝ

to obtain a manifold

N

with boundary. The transformation

F

extends to an automorphism of

N

by setting

F (x, \pm \infty) = (f^{- 1} (x), \pm \infty) .

The

ℤ -

action on

N

generated by

F

is a covering action, and the quotient space

\bar{M_{f}}

is a compact 3-manifold with interior

M_{f}

and boundary

\partial \bar{M_{f}} = \partial^{+} \bar{M_{f}} \cup \partial^{-} \bar{M_{f}},

where

\partial^{\pm} \bar{M_{f}}

is a (possibly disconnected) closed surface homeomorphic to

𝒰^{\pm} / f .

See [21

E. Field, H. Kim, C. Leininger, and M. Loving, End-periodic homeomorphisms and volumes of mapping tori. J. Topol. 16 (2023), no. 1, 57–105 Zbl 1567.57023 MR 4532490

] for a more detailed discussion of the construction. In particular, [21

E. Field, H. Kim, C. Leininger, and M. Loving, End-periodic homeomorphisms and volumes of mapping tori. J. Topol. 16 (2023), no. 1, 57–105 Zbl 1567.57023 MR 4532490

, Lemma 3.3] shows that

\bar{M_{f}}

is atoroidal if and only if

f

is atoroidal.

2.3. Depth-one foliation

A foliation

ℱ

in

M

is a depth-one foliation if

ℱ

has finitely many compact leaves, whose union we denote by

ℱ^{0},

and

ℱ

restricted to

M - ℱ^{0}

is a fibration over a circle with non-compact fibers. A connected component of

M - ℱ^{0}

is called a fibered region of

M .

Any fibered region

Ω

is bounded by leaves in

ℱ^{0},

and we denote the collection of these leaves by

\partial Ω .

We say a leaf in

\partial Ω

is a positive (resp. negative) boundary leaf of

Ω

if it is on the

ϕ -

positive (resp.

ϕ -

negative) side of

Ω .

We denote the collection of positive/negative boundary leaves by

\partial^{\pm} Ω .

Note that it is possible to have a compact leaf contained in both

\partial^{+} Ω

and

\partial^{-} Ω .

Let

\bar{Ω}

be the union of

Ω

and

\partial Ω .

Let

L

be a fiber of the fibration

{ℱ |}_{Ω} .

The leaf

L

limits on a compact leaf

Σ \subset \partial Ω

in the following way [9

J. Cantwell and L. Conlon, Poincaré–Bendixson theory for leaves of codimension one. Trans. Amer. Math. Soc. 265 (1981), no. 1, 181–209 Zbl 0484.57015 MR 0607116

]. Let

N (Σ) ≅ Σ \times [- 1, 1]

be a regular neighborhood of

Σ

with

Σ

identified with

Σ \times {0},

and assume that

L

limits on

Σ

on the positive side. If

N (Σ)

is small enough, the intersection of

L

and

N (Σ)

is an infinite surface spiraling to

Σ

and covering

Σ

with infinite degree. More precisely, up to shrinking

N (Σ),

there is a multi-curve

δ

on

Σ

so that

L \cap N (Σ)

is isotopic to an oriented cut-and-paste of

δ \times (0, 1]

and

⋃_{n \geq 2} Σ \times {1 / n} .

A fundamental domain for the spiraling is depicted on top of Figure 1a, and a schematic picture of the spiraling neighborhood is shown in Figure 1b.

We say a foliation

ℱ

is taut if for any leaf of

ℱ,

there is a transverse loop intersecting that leaf.

Figure 1.

A schematic picture of the spiraling neighborhood of

Σ .

Convention 2.5.

We fix a taut depth-one foliation

ℱ

of

M

and assume that

ℱ

is transverse to

ϕ .

By transversality,

ℱ

is coorientable, and we take the orientation to be consistent with the flow direction.

By transversality of

ℱ

to

ϕ

and by the compactness of

M,

the angle between

T ϕ

and

T ℱ

at any point is uniformly bounded away from zero. In particular, this implies that

{ϕ |}_{Ω}

is a suspension flow of the fibration

{ℱ |}_{Ω} .

The flow gives us a way to identify each

ℱ -

leaf in

Ω

with

L .

Since

ℱ

is taut, every leaf in

ℱ^{0}

is homologically non-trivial and incompressible by Novikov [29

S. P. Novikov, The topology of foliations (in Russian). Tr. Moskov. Mat. Obs. 14 (1965), 248–278. English translation. Trans. Moscow Math. Soc. 14 (1965), 268–304 Zbl 0247.57006 MR 0200938

]. Since

M

is atoroidal, every compact leaf is a closed hyperbolic surface (the possibility of being a sphere is ruled out by the Reeb stability theorem). Therefore, the fundamental domains of the spiraling of

L

can be chosen to be non-planar. In a spiraling neighborhood of a compact leaf,

ϕ

looks like the product flow. One can see that the first return map induced by

ϕ

is an end-periodic homeomorphism

f : L \to L

[15

S. R. Fenley, Asymptotic properties of depth one foliations in hyperbolic

3 -

manifolds. J. Differential Geom. 36 (1992), no. 2, 269–313 Zbl 0766.53018 MR 1180384

]. Since

M

is atoroidal, so is

f .

The metric completion of

Ω

with respect to the path metric induced by the metric in

M

gives a compactified mapping torus of

f .

By [8

A. Candel, Uniformization of surface laminations. Ann. Sci. Éc. Norm. Super. (4) 26 (1993), no. 4, 489–516 Zbl 0785.57009 MR 1235439

], there is a Riemannian metric on

M

that restricts to hyperbolic metrics on leaves of

ℱ .

We fix such a metric on

M .

A hyperbolic metric on a surface is standard if there is no embedded half-space, following [10

J. Cantwell and L. Conlon, Hyperbolic geometry and homotopic homeomorphisms of surfaces. Geom. Dedicata 177 (2015), no. 1, 27–42 Zbl 1359.37097 MR 3370020

]. The induced hyperbolic metric on any depth-one leaf of

ℱ

has injectivity radius bounded from above. This is discussed in [23

M. P. Landry, Y. N. Minsky, and S. J. Taylor, Endperiodic maps via pseudo-Anosov flows. 2023, arXiv:2304.10620v1, to appear in Geom. Topol.

, Proof of Proposition 4.6], for instance, and the rough idea is that the junctures on any depth-one leaf have bounded length, and any point on the leaf is at a bounded distance from a juncture.

Let

𝒰^{\pm} \subset L

be the positive or negative escaping set of

f .

By definition, a point

x \in L

is in

𝒰^{+}

if and only if the positive ray of

ϕ (x)

hits

\partial^{+} Ω,

and it is in

𝒰^{-}

if and only if the negative ray of

ϕ (x)

hits

\partial^{-} Ω .

If

(\bar{Ω}, \partial^{-} Ω, \partial^{+} Ω) ≅ (S \times [0, 1], S \times {0}, S \times {1})

or

(S \times S^{1}, S \times {1}, S \times {1})

for some closed hyperbolic surface

S,

we say

Ω

is a trivial fibered region. This is equivalent to the monodromy

f

being a pure translation on

L

and to

𝒰^{-} = 𝒰^{+} = L

[11

J. Cantwell, L. Conlon, and S. R. Fenley, Endperiodic automorphisms of surfaces and foliations. Ergodic Theory Dynam. Systems 41 (2021), no. 1, 66–212 Zbl 1464.57022 MR 4190052

, Proposition 4.76].

Convention 2.6.

We assume that

M

has no trivial product region. In particular, any compact leaf of

ℱ

is not a fiber. All of our discussions and statements hold with trivial product regions and are readily checked, so we omit the related discussion for simplicity.

Let

\tilde{ℱ}

be the lift of

ℱ

in

\tilde{M} .

The lifted foliation

\tilde{ℱ}

is a foliation by planes by [29

S. P. Novikov, The topology of foliations (in Russian). Tr. Moskov. Mat. Obs. 14 (1965), 248–278. English translation. Trans. Moscow Math. Soc. 14 (1965), 268–304 Zbl 0247.57006 MR 0200938

]. A connected lift of a fibered region of

M

is called a product region of

\tilde{M} .

Any product region

\tilde{Ω}

covering

Ω

is homeomorphic to

\tilde{L} \times ℝ,

and we fix a homeomorphism so that

\tilde{Ω}

is foliated by

\tilde{L} \times {t}

and the

\tilde{ϕ} -

orbits are

{x} \times ℝ .

A lift of a positive/negative boundary leaf of

Ω

is a positive/negative boundary leaf of

\tilde{Ω},

the collection of which is denoted by

\partial^{\pm} \tilde{Ω} .

Define

\partial \tilde{Ω} := \partial^{+} \tilde{Ω} \cup \partial^{-} \tilde{Ω} and \bar{\tilde{Ω}} := \tilde{Ω} \cup \partial \tilde{Ω} .

Let

{\tilde{𝒰}}^{\pm}

be the preimage of

𝒰^{\pm}

in

\tilde{L} .

From the construction of the compactified mapping torus, we see that

\bar{\tilde{Ω}}

is homeomorphic to

(2.1)

(\tilde{L} \times ℝ) \cup ({\tilde{𝒰}}^{+} \times {+ \infty}) \cup ({\tilde{𝒰}}^{-} \times {- \infty}),

where

{\tilde{𝒰}}^{\pm}

is the preimage of

𝒰^{\pm}

in

\tilde{L} .

For any leaf

μ \in \partial^{+} \tilde{Ω},

there is a component

{\tilde{𝒰}}_{μ}^{+}

of

{\tilde{𝒰}}^{+}

such that a

\tilde{ϕ} -

orbit intersects

μ

if and only if it intersects

\tilde{L}

at a point contained in

{\tilde{𝒰}}_{μ}^{+} .

This gives a bijection between leaves in

\partial^{+} \tilde{Ω}

and components of

{\tilde{𝒰}}^{+} .

Similarly, there is a bijection between leaves in

\partial^{-} \tilde{Ω}

and components of

{\tilde{𝒰}}^{-} .

A leaf of

\tilde{ℱ}

is called a type-0 leaf if it covers a leaf in

ℱ^{0} .

Otherwise, we call it a type-1 leaf. Every type-1 leaf

μ

is contained in a unique product region in

\tilde{M},

denoted by

\tilde{Ω} (μ) .

Every type-0 leaf is the negative boundary of a unique production region, and the positive boundary of another different product region. A type-0 leaf

λ

and a type-1 leaf

μ

are called adjacent if

λ \subset \partial \tilde{Ω} (μ) .

If moreover

λ

is in the positive side of

μ,

we say

λ

is positively adjacent to

μ

or

μ ≲ λ;

otherwise, we say

λ

is negatively adjacent to

μ

or

μ ≳ λ .

Let

Λ

be the leaf space of

\tilde{ℱ}

obtained by collapsing each leaf to a point, which is a non-Hausdorff 1-manifold. Since each leaf of

\tilde{ℱ}

is properly embedded in

\tilde{M}

and hence separating,

Λ

is simply connected. Each product region

\tilde{Ω}

projects to an oriented open interval in

Λ,

with the orientation induced by

\tilde{ϕ} .

Every such open interval has a countably infinite number of positive endpoints, each corresponding to a component of

\partial^{+} \tilde{Ω} .

See Figure 2 for an illustration of how leaves in the product regions in

\tilde{M}

limit to different type-0 leaves. The positive endpoints of the open intervals are non-separated from each other in

Λ

by (2.1). The same is true for negative endpoints. The closures of product regions are glued together along type-0 leaves in the boundary, so each point corresponding to a type-0 leaf is the negative endpoint of exactly one open interval associated to a product region, and the positive endpoint of exactly another different one.

Figure 2.

Type-1 leaves (black) in a product region limit to type-0 leaves (blue) in the positive boundary and stay transverse to

\tilde{ϕ}

(red), creating non-Hausdorffness in

Λ .

Figure 3.

Left: a local picture near a product region

\tilde{Ω}

in

Λ .

Right: the corresponding parts in

Λ^{*} .

The red vertices represent type-0 leaves in both pictures, and the arrows indicate the direction of

\tilde{ϕ} .

It is sometimes useful to think about the dual graph

Λ^{*} .

The set of vertices is the set of product regions and type-0 leaves. For a product region or a type-0 leaf

x,

we denote the dual vertex in

Λ^{*}

by

x^{*} .

The edges are the pairs

(Ω^{*}, μ^{*})

where

Ω

is a product region,

μ

is a type-0 leaf and

μ \in \partial Ω .

The dual graph

Λ^{*}

is an infinite valence tree with an orientation given by the flow (see Figure 3).

Two leaves of

\tilde{ℱ}

are called comparable if they can be connected by an oriented path in

Λ .

Otherwise, they are incomparable. If

λ

is comparable to

μ

and

μ

is on the positive side of

λ,

we write

λ < μ

or

μ > λ .

Similarly, we say two vertices are comparable if there is an oriented path connecting them in

Λ^{*}

and are incomparable if otherwise. If two vertices

v

and

w

are comparable and

v

is on the positive side of

w,

we write

w < v

or

v > w .

Two product regions are comparable/incomparable if their dual vertices are comparable/incomparable.

2.4. Laminations on $S^{1}$

We recall some definitions and constructions of abstract laminations on

S^{1} .

Let

{Symm}_{2} (S^{1}) := S^{1} \times S^{1} - Δ / \sim

be the space of unordered pairs of distinct points in

S^{1}

endowed with the quotient topology, where the relation is given by

(x, y) \sim (y, x) .

Two pairs of points

{x, y}

and

{z, w}

on

S^{1}

are said to be unlinked if

z

and

w

lie in the same component of

S^{1} - {x, y} .

A lamination on

S^{1}

is a closed pairwise unlinked subset of

{Symm}_{2} (S^{1}) .

By identifying

S^{1}

with

\partial ℍ^{2},

any lamination

Ξ

on

S^{1}

determines a geodesic lamination

Ξ_{geod}

on

ℍ^{2}

by taking the union of geodesics that connect the pairs in

Ξ .

Conversely, any geodesic lamination in

ℍ^{2}

gives a lamination on

S^{1}

consisting of endpoint pairs of leaves.

Given any subset

A \subset S^{1},

the boundary of the convex hull of

\bar{A}

is a geodesic laminations on

ℍ^{2},

which can be viewed as a lamination

\partial CH (A)

on

S^{1} .

Note that the lamination

\partial CH (A)

is independent of the choice of the identification

S^{1} ≅ \partial ℍ^{2} .

3. Infinity of shadows

Let

p : \tilde{M} \to 𝒪

be the projection map. For any subset

A

of

\tilde{M},

the image of

A

under

p

is called the shadow of

A .

In this section, we will study the shadow of leaves of

\tilde{ℱ},

especially the behavior of the shadow at infinity. The main results are Lemmas 3.3 and 3.5, which hint at a universal circle structure on

\partial 𝒪 .

Let

λ

be a type-0 leaf of

\tilde{ℱ},

and let

Σ

be the leaf of

ℱ

it covers. Denote the covering map by

π_{0} .

Since

λ

separates

\tilde{M}

and the flow

\tilde{ϕ}

crosses

λ

positively, each flowline intersects

λ

at most once. It follows that

p

restricted to

λ

is a continuous bijection to its image. This map

{p |}_{λ} : λ \to p (λ)

also has a continuous inverse, mapping a point in

p (λ)

to the intersection of the corresponding orbit with

λ .

Therefore,

{p |}_{λ}

is a homeomorphism to

p (λ) .

The map

π := π_{0} \circ {({p |}_{λ})}^{- 1} : p (λ) \to Σ

is then a covering map, so we can view

p (λ)

as a universal cover of

Σ .

The deck transformation group action on

p (λ)

is simply the action of

π_{1} (M)

on

𝒪

restricted to a subgroup in the conjugacy class of

π_{1} (Σ)

that stabilizes

p (λ) .

We identify this subgroup with

π_{1} (Σ) .

By transversality, the intersection of

ℱ^{s / u}

with

Σ

induces a singular foliation on

Σ,

denoted by

ℱ_{Σ}^{s / u} .

Similarly, let the intersection of

{\tilde{ℱ}}^{s / u}

with

λ

be denoted by

{\tilde{ℱ}}_{λ}^{s / u} .

These foliations are related by the relations

p ({\tilde{ℱ}}_{λ}^{s / u}) = {ℱ_{𝒪}^{s / u} |}_{p (λ)}

and

π ({ℱ_{𝒪}^{s / u} |}_{p (λ)}) = ℱ_{Σ}^{s / u} .

Shadows of type-0 leaves. Assume

λ

is a type-0 leaf, so

Σ

is an embedded closed surface in

M

which is not a fiber by Convention 2.6. The shape of the shadow of a non-fibered transverse closed embedded surface is carefully studied in [12

D. Cooper, D. D. Long, and A. W. Reid, Bundles and finite foliations. Invent. Math. 118 (1994), no. 1, 255–283 Zbl 0858.57015 MR 1292113

] when

ϕ

is a pseudo-Anosov suspension flow, later generalized by [19

S. R. Fenley, Surfaces transverse to pseudo-Anosov flows and virtual fibers in

3 -

manifolds. Topology 38 (1999), no. 4, 823–859 Zbl 0926.57009 MR 1679801

] to general pseudo-Anosov flows. Each component of the topological boundary of

p (λ)

in

𝒪

is either a regular leaf of

ℱ_{𝒪}^{u}

or

ℱ_{𝒪}^{s}

or a face of a singular leaf that is regular on the side containing

p (λ)

[19

S. R. Fenley, Surfaces transverse to pseudo-Anosov flows and virtual fibers in

3 -

manifolds. Topology 38 (1999), no. 4, 823–859 Zbl 0926.57009 MR 1679801

, Proposition 4.3]. We call a boundary component of

p (λ)

in

𝒪

a side of

p (λ) .

Consider a side

e

of

p (λ),

which we assume to be contained in a leaf of

ℱ_{𝒪}^{s}

for concreteness. We collect some useful facts about the local dynamics at

e

from [19

S. R. Fenley, Surfaces transverse to pseudo-Anosov flows and virtual fibers in

3 -

manifolds. Topology 38 (1999), no. 4, 823–859 Zbl 0926.57009 MR 1679801

] in the following proposition.

Proposition 3.1

([19

S. R. Fenley, Surfaces transverse to pseudo-Anosov flows and virtual fibers in

3 -

manifolds. Topology 38 (1999), no. 4, 823–859 Zbl 0926.57009 MR 1679801

]). The stabilizer of $e$ in $π_{1} (M)$ is isomorphic to $ℤ$ and contained in $π_{1} (Σ) .$ There is a generator $g_{e}$ acting on $p (λ)$ with the following dynamics (Figure 4):

The element $g_{e}$ acts as a contraction on $e$ with a unique fixed point $x_{e}$ .
The element $g_{e}$ fixes and expands $l_{e} := ℱ_{𝒪}^{u} (x_{e}) \cap p (λ)$ , which is the interior of a ray of $ℱ_{𝒪}^{u}$ . We have that $l_{e}$ projects via $π$ to a closed leaf $α_{e}$ of $ℱ_{Σ}^{u}$ , whose free homotopy class is represented by $g_{e}$ .
For any point $x$ other than $x_{e}$ in $e$ , the intersection $l_{x} := ℱ_{𝒪}^{u} (x) \cap p (λ)$ is connected, and it projects via $π$ to a non-compact leaf of $ℱ_{Σ}^{u}$ that spirals into $α_{e}$ . If we orient $l_{x}$ and $l_{e}$ so that they point toward $e$ and let $π (l_{x})$ and $α_{e}$ inherit the orientation, then $π (l_{x})$ and $α_{e}$ are asymptotic in the forward direction.

Figure 4.

The dynamics near a side of the shadow of a type-0 leaf.

The 3-dimensional picture is the following (Figure 5). In

\tilde{M},

the orbit

σ_{e} := p^{- 1} (x_{e})

is a periodic orbit disjoint from

λ .

The leaf

p^{- 1} (e) \subset {\tilde{ℱ}}^{s} (σ_{e})

does not intersect

λ,

and

p^{- 1} (l_{e}) \subset {\tilde{ℱ}}^{u} (σ_{e})

intersects

λ

transversely in a line

{\tilde{α}}_{e}

so that all the

\tilde{ϕ} -

orbits in

p^{- 1} (l_{e})

cross

λ

positively. The line

{\tilde{α}}_{e}

covers the simple closed curve

α_{e}

in

Σ .

The element

g_{e}

is a translation that fixes

σ_{e}

and

λ .

For a side

e

contained in a leaf of

ℱ_{𝒪}^{u},

we have a similar picture. From now on, given a side

e

of

p (λ),

we will continue to use the notations

g_{e},

l_{e},

x_{e}

and

α_{e}

for the objects described in Proposition 3.1. In particular, we always take

g_{e}

to be the generator in the stabilizer of

e

that contracts

e .

Figure 5.

The vertical arrowed line represents the periodic orbit

σ_{e}

(assumed to be regular), the red plane is

{\tilde{ℱ}}^{s} (σ_{e}),

the blue plane is

{\tilde{ℱ}}^{u} (σ_{e})

and the green plane is

λ .

Let

\partial p (λ)

be the boundary of

p (λ)

in $\bar{𝒪}$ , consisting of sides and ideal boundary points at infinity of

p (λ) .

Lemma 3.2.

The intersection $\partial p (λ) \cap \partial 𝒪$ is nowhere dense.

Proof.

First we observe that

\partial p (λ) \cap \partial 𝒪

is a closed subset of

\partial 𝒪 .

This is because the complement of

\partial p (λ) \cap \partial 𝒪

on

\partial 𝒪

is a union of disjoint open intervals bounded by the endpoints of sides of

p (λ) .

Suppose that there is a maximal closed interval

A

contained in

\partial p (λ) \cap \partial 𝒪

with endpoints

η_{1}

and

η_{2} .

Then there is a side

e_{1}

of

p (λ)

such that

η_{1}

is an endpoint of

e_{1} .

Take

g_{e_{1}},

x_{e_{1}}

and

l_{e_{1}}

as before. Let

c

be the endpoint of

l_{e_{1}}

at

\partial 𝒪 .

There is another side

e_{2}

of

p (λ)

such that

η_{2}

is an endpoint of

e_{2} .

If

c

does not lie in

A,

then

e_{2}

is between

c

and

η_{1}

(Figure 6). The action of

g_{e_{1}}

fixes

η_{1}

and

c,

but it cannot fix

e_{2} .

Otherwise,

g_{e_{1}}

will have two fixed points on

𝒪

by Lemma 2.2, contradicting the assumption that

ϕ

has no perfect fits by Lemma 2.3. Then one of

g_{e_{1}}^{\pm 1} (η_{2})

will be in

A,

contradicting our choice of

A .

Figure 6.

Proof of Lemma 3.2.

Hence,

c

must lie in

A .

Now

l_{e_{1}}

divides

p (λ)

into two connected components, one of which contains no side of

p (λ) .

In particular, this component contains no

π_{1} (Σ) -

translation of

l_{e_{1}} .

Translated to the hyperbolic plane by

Q_{λ},

this means there is a simple closed curve

α_{e_{1}} \in Σ

and a lift

Q_{λ} (l_{e_{1}})

of

α_{e_{1}}

in

λ

such that there is no other lift on one side of

\tilde{α} .

But that is impossible. We conclude that such an interval

A

does not exist.

The boundary

\partial p (λ)

is homeomorphic to a circle, and

\bar{p (λ)} := \partial p (λ) \cup p (λ)

is the closure of

p (λ)

in

\bar{𝒪} .

Note that our choice of the metric on

M

restricts to a hyperbolic metric on

Σ,

so

λ

is isometric to

ℍ^{2} .

We define

\bar{λ}

to be the usual compactified hyperbolic plane with ideal boundary

\partial_{\infty} λ .

The next lemma reveals how

\partial p (λ)

is related to

\partial_{\infty} λ .

A map

g

from

S^{1}

to

S^{1}

is monotone if the preimage of any point on

S^{1}

is contractible. A gap of

g

is a maximal closed interval of positive length in

S^{1}

that is collapsed to a single point by

g .

The core of

g

is the complement of the union of the interiors of gaps, denoted by

core (g) .

We remark that the gaps are sometimes taken to be open intervals in the literature, different from our convention.

Lemma 3.3.

Let $λ$ be a type- $0$ leaf of $\tilde{ℱ} .$ Then the homeomorphism ${p^{- 1} |}_{p (λ)} : p (λ) \to λ$ extends continuously to a map $Q_{λ}$ from $\bar{p (λ)}$ to $\bar{λ} := \partial_{\infty} λ \cup λ .$ The map restricted to $\partial p (λ)$ is a monotone map to $\partial_{\infty} λ$ with core $\bar{p (λ)} \cap \partial 𝒪 .$

Proof.

To begin with, we define

Q_{λ} = {p^{- 1} |}_{p (λ)}

in the interior of

p (λ) .

To extend

Q_{λ}

to

\partial p (λ),

we will first define the map on the sides of

p (λ)

and then extend it to the entire

\partial p (λ) .

Let

e

be a side of

p (λ) .

Take

g_{e} \in Stab (e),

x_{e} \in e

and

l_{e}

as in Proposition 3.1. As an element of

π_{1} (Σ),

g_{e}

acts as a hyperbolic element on

\bar{λ}

with a contracting fixed point

\partial^{-} g_{e}

and a repelling fixed point

\partial^{+} g_{e}

at infinity. We define

Q_{λ}

on

e

as the constant map to

\partial^{-} g_{e} .

Different sides are sent to different points in

\partial_{\infty} (λ)

because of Lemma 2.3.

Since

Q_{λ} (l_{e})

projects to a simple closed curve

α_{e}

in

Σ,

it is a quasi-geodesic in

λ

with well-defined endpoints in

\partial_{\infty} λ .

Moreover,

α_{e}

represents the free homotopy class of

g_{e}

up to taking the inverse, by Proposition 3.1. Therefore, the endpoints of

Q_{λ} (l_{e})

are

\partial^{\pm} g_{e} .

By the way we choose

g_{e}

(item (2) of Proposition 3.1), it contracts

l_{e}

near

e .

Therefore, if we orient

l_{e}

to point toward

e

and give

Q_{λ} (l_{e})

the induced orientation, the forward endpoint of

Q_{λ} (l_{e})

is

\partial^{-} g_{e} .

This shows

Q_{λ}

is continuous at

x_{e}

when restricted to

l_{e} \cup x_{e} .

This further ensures that

Q_{λ}

preserves the cyclic order of the boundary leaves, which is a consequence of the fact that

Q_{λ}

is an orientation-preserving homeomorphism in the interior. For example, one can argue as follows (indicated in Figure 7). Consider three sides

e_{1}, e_{2}

and

e_{3}

in clockwise order, and take

x_{e_{i}}

and

l_{e_{i}}

as before. Take a point

y_{e_{i}} \in l_{e_{i}},

and let

l_{e_{i}}^{'} \subset l_{e_{i}}

be the segment between

x_{e_{i}}

and

y_{e_{i}} .

We may ensure

l_{e_{i}}^{'}

are pairwise disjoint by taking

y_{e_{i}}

close enough to

x_{e_{i}} .

Take any point

z \in p (λ)

that is not in the union of

l_{e_{i}}^{'},

and connect

y_{e_{i}}

to

z

to get an embedded 3-prong

P

with

z

as the center and

x_{e_{i}}

as the endpoints. The image

Q_{λ} (P)

is an embedded 3-prong in

\bar{λ}

with well-defined endpoints

Q_{λ} (e_{i}) .

The cyclic order of the edges of

R

is preserved under

Q_{λ},

assuring that

Q_{λ} (e_{1}), Q_{λ} (e_{2})

and

Q_{λ} (e_{3})

are arranged clockwise.

Figure 7.

Q_{λ}

preserves the cyclic order.

Moreover, if

γ

is an element of

π_{1} (Σ),

it translates

e

to another side

γ e

of

p (λ) .

We have

g_{γ e} = γ g_{e} γ^{- 1},

so

Q_{λ} (γ e) = γ Q_{λ} (e) .

By the minimality of the

π_{1} (Σ)

action on

\partial_{\infty} λ,

the

Q_{λ} -

images of all the sides of

p (λ)

are dense in

\partial_{\infty} λ .

Now there is a unique way to define

Q_{λ}

on

\partial p (λ)

such that it is continuous when restricted to

\partial p (λ) .

That is, for any point

x \in \partial p (λ),

one can find a sequence of sides

e_{n}

converging to

x

and define

Q_{λ} (x)

as the limit of

Q_{λ} (e_{n})

by Lemma 3.2. The limit exists and is independent of

e_{n}

because

Q_{λ}

preserves the cyclic order of sides and the image of sides is dense in

\partial_{\infty} λ .

To complete the proof of Lemma 3.3, what is left is to check that

Q_{λ}

is continuous on

\bar{p (λ)} .

For a side

e

(which is assumed to be contained in a leaf of

ℱ_{𝒪}^{s}

for concreteness) of

p (λ)

and a point

x \in e,

we take a rectangular neighborhood of

x

as the image of an embedding

ρ : (0, 1) \times [0, 1) \to \bar{p (λ)}

such that

$ρ (1 / 2, 0) = x;$
$ρ ((0, 1) \times {0})$ is contained in $e;$
for any $s \in [0, 1),$ $ρ ((0, 1) \times {s})$ is contained in a leaf of $ℱ_{𝒪}^{s};$
for any $t \in (0, 1),$ $ρ ({t} \times (0, 1))$ is contained in a leaf of $ℱ_{𝒪}^{u} .$

Such a neighborhood always exists because

ℱ_{𝒪}^{s}

is regular on the side of

e

that contains

p (λ) .

Fix a rectangular neighborhood

RN (x)

of

x .

By Proposition 3.1, we know that

Q_{λ} (ρ ({t} \times (0, 1)))

is asymptotic to

Q_{λ} (l_{e}),

the latter being a quasi-geodesic ray in

λ

since it is a lift of a simple closed curve in

Σ .

This shows that

Q_{λ} (RN (x))

is a wedge-shaped region in

λ

with one ideal point

Q_{λ} (e)

(Figure 8).

Figure 8.

The

Q_{λ} -

image of

ℱ_{𝒪}^{s} -

leaves near an unstable boundary leaf

e .

Consider a sequence of points

x_{n}

in

p (λ)

converging to

x \in \partial p (λ) .

If

x

is on a side

e,

we can take a rectangular neighborhood of

x

that will eventually contain

x_{n} .

The shape of the rectangular neighborhoods under

Q_{λ}

shows that

Q_{λ} (x_{n})

converges to

Q_{λ} (x) = Q_{λ} (e) .

Figure 9.

Continuity of

Q_{λ}

at

\partial p (λ) \cap \partial 𝒪 .

If

x \in \partial p (λ) \cap \partial 𝒪,

we can trap

Q_{λ} (x_{n})

using segments of

l_{e},

forcing them to converge to the right points. The argument is similar to the one we use to prove that

Q_{λ}

preserves the cyclic order of the sides. See Figure 9 for an illustration. More precisely, take sequences of sides

{e_{m}^{+}}

and

{e_{m}^{-}}

of

p (λ)

that approximate

x

from two sides respectively (using Lemma 3.2). Fixing

m,

take a short segment

l_{e_{m}^{\pm}}^{'} \subset l_{e_{m}^{\pm}}

with one endpoint at

x_{e_{m}^{\pm}}

so that

l_{e_{m}^{+}}^{'}

and

l_{e_{m}^{-}}^{'}

are disjoint. Connect the other endpoints of the two segments by a path in

p (λ),

and denote the resulting path by

α_{m} .

The image

Q_{λ} (α_{m})

is an embedded line in

\tilde{λ}

with disjoint endpoints, separating

\tilde{λ}

into two half-planes. We let the half-plane containing

Q_{λ} (x)

be

H_{m} (x) .

The fact that

{Q_{λ} |}_{p (λ)}

is an orientation-preserving homeomorphism guarantees that

Q_{λ} (x_{n})

eventually enters

H_{m} (x)

for all

m .

We can arrange

H_{m} (x)

to be nested so that

⋂_{m} H_{m} (x)

has exactly one ideal point

Q_{λ} (x) .

Since

{x_{n}},

hence

{Q_{λ} (x_{n})},

escapes every compact set, we know that

{Q_{λ} (x_{n})}

must limit to

Q_{λ} (x) .

We have proved that

Q_{λ}

is continuous on

\bar{p (λ)},

finishing the proof of Lemma 3.3.

Shadows of type-1 leaves. In this subsection, we would like to study the shadow of type-1 leaves in

\tilde{ℱ} .

Note that by [20

S. R. Fenley, Geometry of foliations and flows I: almost transverse pseudo-Anosov flows and asymptotic behavior of foliations. J. Differential Geom. 81 (2009), no. 1, 1–89 Zbl 1160.57026 MR 2477891

, Proposition 4.1], the shadow of a leaf of a transverse foliation is always bounded by regular leaves or faces of singular leaves of the stable/unstable foliations. These boundary leaves are in general not periodic for leaves in an arbitrary transverse foliation. However, we will show in the next lemma that they are in fact periodic for type-1 leaves in a depth-one foliation.

We now assume

λ

is a type-1 leaf of

\tilde{F}

contained in a product region

\tilde{Ω} ≅ \tilde{L} \times ℝ

and

λ

covers a non-compact leaf identified with

L .

Similar to the case of type-0 leaves, we will use

\partial p (λ)

to denote the boundary of

p (λ)

in

\bar{𝒪} .

Lemma 3.4.

The shadow $p (λ)$ is a proper open subset of $𝒪,$ bounded by periodic regular leaves of $ℱ_{𝒪}^{s}$ or $ℱ_{𝒪}^{u}$ or faces of singular leaves which are regular on the side containing $p (λ) .$ We call a component of the boundary of $p (λ)$ (as a subset of $𝒪$ ) a side of $p (λ) .$

Moreover, if a side $e$ of $p (λ)$ is contained in a leaf of $ℱ_{𝒪}^{s},$ then there is a type-0 leaf $μ$ negatively adjacent to $λ$ (see Section 2.3 for the definition) such that $e$ is also a side of $p (μ) .$ If $e$ is contained in a leaf of $ℱ_{𝒪}^{u},$ then there is a type- $0$ leaf $μ$ positively adjacent to $λ$ such that $e$ is also a side of $p (μ) .$

Proof.

Suppose

z \in 𝒪

is a boundary point of

p (λ) .

Let

σ_{z} = p^{- 1} (z)

be the

\tilde{ϕ} -

orbit that projects to

z .

The orbit

σ_{z}

cannot intersect any product region that is incomparable to

\tilde{Ω} (λ),

which is the product region containing

λ .

This is because if

σ_{z}

intersects such a product region

\tilde{Ω},

z

will have a neighborhood

U

such that every orbit that projects to

U

also intersects

\tilde{Ω} .

This forces them to be disjoint from

\tilde{Ω} (λ) .

Then

U

is not in

p (λ),

and

z

is not in the boundary of

p (λ) .

Similarly, it cannot intersect any type-0 leaf incomparable to

\tilde{Ω} (λ)

either.

Since

p (λ)

is open,

z

is not contained in

p (λ) .

The orbit

σ_{z}

induces an oriented path

γ_{z}

in the dual graph

Λ^{*}

consisting of all the type-0 leaves and product regions that

σ_{z}

travels through (Section 2). Every vertex in

γ_{z}

is comparable to the dual vertex

\tilde{Ω} {(λ)}^{*}

by the previous paragraph, but

\tilde{Ω} {(λ)}^{*}

is not in

γ_{z} .

Take any vertex

v

in

γ_{z},

and suppose

v > \tilde{Ω} {(λ)}^{*} .

Since

Λ^{*}

is a tree, any vertex

w

satisfying

v > w > \tilde{Ω} {(λ)}^{*}

lies in the unique oriented interval from

v

to

\tilde{Ω} {(λ)}^{*} .

So the path

γ_{z}

has to stay in the interval in the negative direction. The case where

v < \tilde{Ω} {(λ)}^{*}

is similar, and we conclude that the orbit

σ_{z}

will eventually stay in one product region in either the positive or the negative direction.

If

γ_{z}

is on the negative side of

\tilde{Ω} {(λ)}^{*},

then

σ_{z}

eventually stays in a product region, denoted by

{\tilde{Ω}}_{0},

in the positive direction. Take the unique shortest oriented path

γ

from

{\tilde{Ω}}_{0}^{*}

to

\tilde{Ω} {(λ)}^{*}

in

Λ^{*},

and let

{\tilde{Ω}}_{1}^{*}

be the last vertex in

γ

before

\tilde{Ω} {(λ)}^{*} .

There is a unique type-0 leaf

μ

that is positively adjacent to

{\tilde{Ω}}_{1}

and negatively adjacent to

\tilde{Ω} (λ) .

Any

\tilde{ϕ} -

orbit intersecting both

{\tilde{Ω}}_{0}

and

λ

has to enter

\tilde{Ω} (λ)

via

μ .

Conversely, every

\tilde{ϕ} -

orbit intersecting

μ

also meets

λ .

This shows that if an orbit

σ

intersects

{\tilde{Ω}}_{0},

then

p (σ) \in p (μ)

if and only if

p (σ) \in p (λ) .

Since

p ({\tilde{Ω}}_{0})

is open, we have also shown that if

x

is in

p ({\tilde{Ω}}_{0}),

then

x \in \partial p (λ)

if and only if

x \in \partial p (μ) .

Since

z

is in the interior of

p ({\tilde{Ω}}_{0}),

the above reasoning shows that

z

is in the boundary of

p (μ) .

Let

e_{z}

be the side of

p (μ)

containing

z .

Note that all the orbits in the same

{\tilde{ℱ}}^{s} -

leaf as

σ_{z}

are positively asymptotic to

σ_{z} .

In particular, they intersect

{\tilde{Ω}}_{0}

and are disjoint from

μ .

We see that

e_{z}

is contained in a leaf of

ℱ_{𝒪}^{s},

and using a similar argument to the last paragraph, we have

e_{z} \subset \partial p (λ) .

To summarize, we have shown the following: for a point

z \in \partial p (λ),

if

γ_{z}

is on the negative side of

\tilde{Ω} {(λ)}^{*},

then there is type-0 leaf

μ

negatively adjacent to

λ

and a side

e_{z} \in \partial p (μ)

contained in a leaf of

ℱ_{𝒪}^{s}

such that

z \in e_{z}

and

e_{z} \subset \partial p (λ) .

One can apply the same argument to the case where

γ_{z}

is on the positive side of

\tilde{Ω} {(λ)}^{*},

finishing the proof.

Similar to the case of type-0 leaves, we define

\bar{p (λ)}

to be

p (λ) \cup \partial p (λ) .

The following lemma is the type-1 version of Lemma 3.3.

Lemma 3.5.

Let $λ$ be a type- $1$ leaf of $\tilde{ℱ} .$ Then the homeomorphism ${(p_{λ})}^{- 1} : p (λ) \to λ$ extends continuously to a map $Q_{λ}$ from $\bar{p (λ)}$ to $\partial_{\infty} λ .$ The map restricted to $\partial p (λ)$ is a monotone map to $\partial_{\infty} λ$ with core $\bar{p (λ)} \cap \partial 𝒪 .$

Proof.

We define

Q_{λ} = {({p |}_{λ})}^{- 1}

in

p (λ)

and extend

Q_{λ}

to the boundary following the same strategy as in the proof of Lemma 3.3.

Let

e

be any side of

p (λ),

and we assume it to be a leaf of

ℱ_{𝒪}^{s}

for concreteness. By Lemma 3.4,

e

is also a side of the shadow of a type-0 leaf

μ

negatively adjacent to

λ .

Take

g_{e},

x_{e} \in e

and

l_{e}

as in Proposition 3.1. We orient

l_{e}

to be pointing toward

e .

By [20

S. R. Fenley, Geometry of foliations and flows I: almost transverse pseudo-Anosov flows and asymptotic behavior of foliations. J. Differential Geom. 81 (2009), no. 1, 1–89 Zbl 1160.57026 MR 2477891

, Theorem C], each ray of

{\tilde{ℱ}}^{s} \cap λ

or

{\tilde{ℱ}}^{u} \cap λ

has a well-defined endpoint at

\partial_{\infty} λ .

In particular,

Q_{λ} (l_{e})

has a well-defined forward endpoint at infinity. We define

Q_{λ}

on each side

e

of

p (λ)

to be the constant map to the forward endpoint of

Q_{λ} (l_{e}) .

Claim.

Different sides are mapped to different points by

Q_{λ} .

Proof of the claim.

Recall that

\tilde{Ω} (λ)

is the product region containing

λ,

and it covers a fibered region

Ω (λ)

in

M

with fiber

L

and the monodromy

h : L \to L .

Since

M

is atoroidal,

h

is an atoroidal end-periodic map. The fundamental group of

Ω (λ)

is isomorphic to the semidirect product

ℤ ⋉ π_{1} (L)

with

ℤ

acting on

π_{1} (L)

by

h_{*} .

The group

π_{1} (Ω (λ))

stabilizes

p (λ),

and we can define an action of

π_{1} (Ω (λ))

on

λ

by

g (x) := Q_{λ} \circ g \circ p (x)

for

g \in π_{1} (Ω (λ))

and

x \in λ .

If the element

g

has a trivial

ℤ -

factor, then the action of

g

is a covering transformation. Otherwise,

g

acts as a lift of some power of the monodromy

h .

Let

e_{1}

and

e_{2}

be different sides of

p (λ),

and take

g_{e_{i}}

and

x_{e_{i}}

as before for

i = 1, 2 .

By Proposition 3.4, there is a type-0 leaf

μ_{i}

adjacent to

λ

so that

e_{i}

is a side of

p (μ_{1}) .

The type-0 leaf

μ_{i}

covers a compact leaf

Σ_{i} \subset \partial Ω (λ) .

The element

g_{e_{i}},

being a deck transformation for the covering map

μ_{i} \to Σ_{i},

can be viewed as an element of

π_{1} (Ω (λ)) .

Therefore,

g_{e_{i}}

stabilizes

p (λ)

and acts on

λ

either as a hyperbolic isometry or as a lift of some power of

h .

In both cases, the action extends continuously to an automorphism of

\partial_{\infty} λ

by [10

J. Cantwell and L. Conlon, Hyperbolic geometry and homotopic homeomorphisms of surfaces. Geom. Dedicata 177 (2015), no. 1, 27–42 Zbl 1359.37097 MR 3370020

]. Since

g_{e_{i}}

stabilizes

l_{e_{i}}

in

p (λ),

we know

g_{e_{i}}

stabilizes

Q_{λ} (l_{e_{i}})

in

λ,

hence also the point

Q_{λ} (e_{i})

by the definition of

Q_{λ} (e_{i}) .

Suppose for contradiction that

Q_{λ} (e_{1}) = Q_{λ} (e_{2}) = : q .

Then

g_{e_{1}}

and

g_{e_{2}}

fix the same point

q

on

\partial λ .

Note that

g_{e_{1}}

and

g_{e_{2}}

represent different elements in

π_{1} (Ω (λ))

and do not share a non-trivial power by Lemma 2.3. Also note that for any lift

\tilde{h} : λ \to λ

of some power of the monodromy

h,

the action of

\tilde{h}

on

\partial λ

will not fix any fixed point of an element in

π_{1} (L)

that acts as a hyperbolic isometry on

λ .

Otherwise,

h

will fix a closed geodesic

α

on

L

up to isotopy. A regular neighborhood

U (α)

of

α

will also be fixed by

h

up to isotopy, so a power of

h

fixes components of

U (α),

contradicting the assumption that

h

is atoroidal. Therefore, if one of the

g_{e_{i}}

is a hyperbolic isometry, then

g_{e_{1}}

and

g_{e_{2}}

have no common fixed point on

\partial_{\infty} λ .

If both of them are lifts of some power of

h,

up to taking powers we may assume that they are lifts of the same power of

h .

Then there is an element

γ \in π_{1} (λ)

such that

g_{e_{1}} = γ g_{e_{2}} .

The common fixed point

q

will also be fixed by

γ,

a contradiction. We have proved that

g_{e_{1}}

and

g_{e_{2}}

do not have common fixed points, which indicates that

Q_{λ} (e_{1}) \neq Q_{λ} (e_{2}) .

We continue our proof of Lemma 3.5. The map

Q_{λ}

preserves the cyclic order of the sides by the same reason as in the proof of Lemma 3.3, and the image is dense by the minimality of the

π_{1} (L) -

action on

\partial_{\infty} λ .

Indeed, the hyperbolic structure on

L

has bounded injectivity radius, so the limit set of

π_{1} (L)

is the entire circle at infinity.

We also have the following lemma, analogous to Lemma 3.2.

Lemma 3.6.

The intersection

\partial p (λ) \cap \partial 𝒪

is nowhere dense in

\partial 𝒪 .

Proof.

Let

e

be a side of

p (λ) .

By Lemma 3.4,

e

is a side of a shadow of a type-0 leaf. So we can take

g_{e}

as in Proposition 3.1. By the discussion above,

g_{e}

stabilizes

p (λ),

and the action near

e

has the desired expanding-contracting dynamics because of Proposition 3.1. The rest of the proof is the same as that of Lemma 3.2.

To continue the proof of Lemma 3.5, note that by Lemma 3.6, the map

Q_{λ}

can be extended to

\partial p (λ)

by approaching any point in

\partial p (λ) \cap \partial 𝒪

by sides of

\partial p (λ),

as in the proof of Lemma 3.3. The extension is well defined and is continuous on

\partial p (λ) .

To show that the extension is continuous on

p (λ) \cup \partial p (λ),

we again need the nice asymptotic property of rectangular neighborhoods of the sides. What we will show next is basically that Figure 8 is also a correct picture when

λ

is a type-1 leaf.

Again, let

e

be any side of

p (λ),

assumed to be a leaf of

ℱ_{𝒪}^{s} .

Let

μ

be a type-0 leaf negatively adjacent to

λ

such that

e

is also a side of

p (μ),

using Lemma 3.4. Finally, let

g_{e},

x_{e} \in e

and

l_{e}

be as in Proposition 3.1, and orient

l_{e}

to be pointing at

x_{e} .

Let

x^{'} \in e

be a point different from

x_{e}

and set

l^{'}

to be a small segment of

ℱ_{𝒪}^{u} (x^{'}) \cap p (μ)

with an endpoint at

x^{'} .

We orient

l^{'}

to be pointing toward

e

as well. By Proposition 3.1,

Q_{μ} (l^{'})

is forward asymptotic to

Q_{μ} (l_{e})

under the orientation induced from

l^{'}

and

l_{e} .

Since

ℱ_{𝒪}^{s}

is regular near

e

on the side of

p (λ),

there is a pair of points

p_{e} \in l_{e}

and

p^{'} \in l^{'}

so that

p^{'} \in ℱ_{𝒪}^{s} (p_{e}) .

Let

σ_{e}

and

σ^{'}

be the

\tilde{ϕ} -

orbits corresponding to

p_{e}

and

p^{'},

respectively. Since they lie in the same leaf of

{\tilde{ℱ}}^{s},

σ_{e}

and

σ^{'}

are positively asymptotic. It follows that the distance between

σ_{e} \cap λ

and

σ^{'} \cap λ

is bounded by the distance between

σ_{e} \cap μ

and

σ^{'} \cap μ

up to a uniform multiplicative constant. In other words,

d_{λ} (Q_{λ} (p_{e}), Q_{λ} (p^{'}))

is coarsely bounded by

d_{μ} (Q_{μ} (p_{e}), Q_{μ} (p^{'})) .

Moreover, the pair

{p_{e}, p^{'}}

can be chosen arbitrarily close to

e .

Hence,

Q_{λ} (l^{'})

and

Q_{λ} (l_{e})

are forward asymptotic. If

RN (x^{'})

is a rectangular neighborhood of

x^{'}

in

p (μ),

then we have shown that

Q_{λ} (RN (x^{'}))

is a wedge-shaped domain in

λ

meeting

\partial_{\infty} λ

at exactly one point.

Since we have a similar description of the

Q_{λ} -

image of a rectangular neighborhood, we can carry out the same argument and conclude that the extended

Q_{λ}

is continuous on

p (λ) \cup \partial p (λ),

finishing the proof of Lemma 3.5.

Figure 10.

The union of the shaded area is

p (λ),

and the heavily shaded area is

p (μ),

which is a subset of

p (λ) .

The red arrows represent the map

Q_{λ},

and the blue ones represent

Q_{λ μ} .

One should think of each side of

p (λ)

or

p (μ)

as a single point at infinity.

For any leaf

λ

of

\tilde{ℱ},

either type-0 or type-1, we define a map

I_{λ} : \partial 𝒪 \to \partial_{\infty} λ

as follows. For any

ζ \in \partial 𝒪,

if

ζ

is in

\bar{p (λ)},

set

I_{λ} (ζ) = Q_{λ} (ζ) .

If

ζ

is contained in an open interval

V_{ζ}

in

\partial 𝒪 \ \bar{p (λ)},

then there is a boundary leaf

e

of

p (λ)

with the same endpoints as

V_{ζ} .

In this case, we define

I_{λ} (ζ)

to be

Q_{λ} (e) .

It is immediate from the definition that the maps

I_{λ}

are monotone surjections, therefore continuous, and

π_{1} (M) -

equivariant, that is, for any

g \in π_{1} (M),

λ \in \tilde{ℱ}

and

x \in \partial 𝒪,

we have

g I_{λ} (x) = I_{g λ} (g x),

where the action

g : \partial_{\infty} λ \to \partial_{\infty} (g λ)

is induced by the isometry

g : λ \to g λ .

If

λ

is a type-1 leaf and

μ

is a type-0 leaf adjacent to

λ,

then

p (λ)

contains

p (μ),

and the monotone quotient maps

I_{λ}

and

I_{μ}

satisfy the property that for any

ξ_{1}, ξ_{2} \in \partial 𝒪,

I_{λ} (ξ_{1}) = I_{λ} (ξ_{2})

implies

I_{μ} (ξ_{1}) = I_{μ} (ξ_{2}) .

It follows that there is a continuous monotone surjection

I_{λ μ} : \partial_{\infty} λ \to \partial_{\infty} μ

such that

I_{μ} = I_{λ μ} \circ I_{λ} .

More precisely, for any point

ξ \in \partial_{\infty} λ,

I_{λ μ} (ξ)

is defined to be

Q_{μ} (Q_{λ}^{- 1} (ξ)) .

The maps

I_{λ}

and

I_{λ μ}

can be visualized as in Figure 10.

4. Markers and universal circles

The outline of this section is the following. We first recall the definition of a universal circle (Definition 4.1) and prove Theorem 1.1. Then we review in Section 4.1 the construction from [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

] of a particular universal circle

𝔖_{left},

which we call the universal circle from leftmost sections, for any taut foliation. The circle

𝔖_{left}

arises from a collection of special sections, called the leftmost sections, of a circle bundle

E_{\infty}

over

Λ

whose fibers are the circle at infinity of the leaves. The construction will then be examined carefully for our depth-one foliation

ℱ

in Section 4.2, where we study what a leftmost section looks like inside a product region, and in Section 4.3, where we analyze the behavior of a leftmost section at adjacent type-0 and type-1 leaves. The punchlines of this section are Lemmas 4.14 and 4.15, where we show that the leftmost sections can be determined by the structure of

\partial 𝒪

developed in Section 3.

The following axiomatic definition of a universal circle for

ℱ

first appears in [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

]. It is worth remarking that although condition (2) seems not at all natural at first glance, it provides the universal circle with more interesting structures. In particular, it is necessary for the construction of invariant laminations in [5

D. Calegari, Promoting essential laminations. Invent. Math. 166 (2006), no. 3, 583–643 Zbl 1106.57014 MR 2257392

] (cf. Theorem 1.6).

Definition 4.1.

(Universal circle) A universal circle for

ℱ

is a circle

𝔖

with a faithful

π_{1} (M) -

action and a monotone map

U_{λ} : 𝔖 \to \partial_{\infty} λ,

called a structure map, for any leaf

λ

of

\tilde{ℱ}

such that

for any leaf $λ$ and any $γ \in π_{1} (M),$ the following diagram commutes:
if $λ$ and $μ$ are incomparable leaves, then the core of $U_{λ_{1}}$ is contained in a single gap of $U_{λ_{2}},$ and vice versa.

Two universal circles

{𝔖, U_{λ}}

and

{𝔖^{'}, U_{λ}^{'}}

are isomorphic if there is a

π_{1} (M) -

equivariant homeomorphism

h : 𝔖 \to 𝔖^{'}

such that

U_{λ}^{'} \circ h = U_{λ}

for all

λ .

Proof of Theorem 1.1.

All the conditions of a universal circle in Definition 4.1 are obvious by properties of

\partial 𝒪

and the way we define

I_{λ}

except for condition (2). Suppose

λ

and

μ

are incomparable, then their shadows are disjoint. Otherwise, there is an orbit of the flow

\tilde{ϕ}

intersecting both leaves, contradicting their incomparability. Condition (2) is easily seen to be satisfied.

4.1. Calegari–Dunfield’s construction

In [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

], Calegari–Dunfield describe an explicit construction of a universal circle

𝔖_{left}

for any taut foliation. We briefly review their construction below. For simplicity, we will stick to our

ℱ

instead of more general taut foliations.

The bundle

E_{\infty}

is a circle bundle over

Λ

whose fiber at any leaf

λ

is the circle at infinity

\partial_{\infty} λ .

The topology of

E_{\infty}

is defined as follows. For any transversal

τ

of

\tilde{ℱ},

τ

embeds into

Λ,

and we identify

τ

with its embedding image in

Λ .

The unit tangent bundle of

\tilde{ℱ}

restricted to

τ

is the circle bundle

{UT \tilde{ℱ} |}_{τ},

and there is a natural map

{UT \tilde{ℱ} |}_{τ} \to {E_{\infty} |}_{τ}

sending a tangent vector of a leaf to the ideal point it points toward. We require the map to be a homeomorphism. It is shown in [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

] that this topology is well defined, that is, independent of the choice of

τ .

Since

ℱ

is a taut foliation, there is an

ε_{1} > 0

such that every leaf of

\tilde{ℱ}

is quasi-isometrically embedded in its

ε_{1} -

neighborhood by [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

, Lemma 2.4]. By the structure of depth-one foliations, there is a constant

ε_{2} > 0

so that the

ε_{2} -

neighborhood of

ℱ^{0}

is contained in a spiraling neighborhood. Fix

ε_{0}

to be

\min {ε_{1} / 3, ε_{2}} .

Definition 4.2.

A marker for

\tilde{ℱ}

is an embedding

m : [0, 1] \times ℝ_{\geq 0} \to \tilde{M}

such that

for any $x \in [0, 1],$ $m ({x} \times ℝ_{\geq 0})$ is a geodesic ray in a leaf of $\tilde{ℱ};$
for any $y \in ℝ_{+},$ $m ([0, 1] \times {y})$ is a transversal with length bounded by $ε_{0} .$

Any marker

m

gives a section

s

of

{E_{\infty} |}_{τ},

where

τ

is the image of

m ([0, 1] \times {y})

in

Λ,

such that for any leaf

λ \in τ,

s (λ)

is the ideal endpoint of

Image (m) \cap λ .

The image of

τ

under

s

is called the end of

m .

Note that our choice of

ε_{0}

is different from but no larger than the constant chosen in [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

]. Shrinking the constant will not affect the main results in their paper. In general, different

ε

might give rise to different universal circles, but in our case, it can be seen that for

ε_{0}

small enough (i.e., smaller than

ε_{2}

above), the leftmost universal circles are all isomorphic.

Lemma 4.3 ([7
D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363
]).

Given two marker ends, either they are disjoint or their union is an embedded closed interval in $E_{\infty}$ transverse to fibers.

A point

ξ \in \partial λ

is called a marker endpoint if there is a marker

m

so that the end of

m

intersects

\partial_{\infty} λ

at

ξ .

The following theorem was originally announced by Thurston in an unpublished manuscript [32

W. P. Thurston, Three-manifolds, foliations and circles, II. Unfinished manuscript, 1998

], and the proof is carefully written down in [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

]. Heuristically, it says that the leaves of

\tilde{ℱ}

stay close in many directions.

Theorem 4.4.

(Thurston’s leaf pocket theorem) For any leaf $λ$ of $\tilde{ℱ},$ the set of marker endpoints in $\partial_{\infty} λ$ is dense.

For any leaf

λ

of

\tilde{ℱ}

and any point

ξ \in \partial_{\infty} λ,

there is a special section

s_{ξ}

of

E_{\infty},

called the leftmost section starting from

ξ,

built as follows.

In

Λ,

there is a neighborhood

τ

of

λ

homeomorphic to a closed interval, and

E_{\infty} | τ

is a cylinder. We adopt the convention that the flow

\tilde{ϕ}

is flowing upward, and we are facing the cylinder

E_{\infty} | τ

from the outside. Take a finite collection

C

of marker ends in

E_{\infty} | τ

so that each fiber intersects at least one element in

C .

This is possible by Theorem 4.4. We build a path

α_{C}

by starting from

ξ,

heading left horizontally in a fiber until we hit the first marker end in

C

and following the marker end to move upward. After we reach the top of the marker end, we turn left again, staying in a fiber until we hit the next marker end in

C,

and follow the same rules to move on until we reach the top of

{E_{\infty} |}_{τ} .

We call this the leftmost-up rule, following [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

]. We can also move downward from

ξ,

but in the rightmost-down way. This procedure gives us a staircase path

α_{C}

in

{E_{\infty} |}_{τ},

which is an approximation to

s_{ξ}

(Figure 11).

Figure 11.

Approximating the leftmost section on

{E_{\infty} |}_{τ}

by starting from

ξ

and going leftmost up.

To go from the staircase approximations to the leftmost section

s_{ξ},

we define

{s_{ξ} |}_{τ}

to be the (rightmost) supremum above

ξ

and the (leftmost) infimum below

ξ

among all possible

α_{C} .

To be precise, we view

{E_{\infty} |}_{τ}

as

τ \times (ℝ / ℤ) .

For a leaf

μ \in τ

above

λ,

we define

s_{ξ} (μ) = sup_{C} (\min (α_{C} \cap \partial_{\infty} μ)) .

For

μ \in τ

below

λ,

define

s_{ξ} (μ) = inf_{C} (\max (α_{C} \cap \partial_{\infty} μ)) .

It was proved in [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

] that the supremum and the infimum exist, and

s_{ξ}

is indeed a continuous section of

E_{\infty}

over

τ .

We can define

s_{ξ}

for all leaves comparable to

λ

following this procedure.

Finally, we can branch out in

Λ

by turning around to reach incomparable leaves where

s_{ξ}

is not yet defined. More precisely, suppose

μ

is a leaf incomparable to

λ .

There is a sequence of leaves

λ_{0} = λ, λ_{1}, \dots, λ_{n} = μ

so that

λ_{2 i}

and

λ_{2 i + 1}

are comparable and

λ_{2 i + 1}

and

λ_{2 i + 2}

are non-separated. To illustrate the idea, we assume

λ_{1}

is above

λ

(see Figure 12 for the case when

n = 9

). In this case, there is a product region

{\tilde{Ω}}_{1}

so that

λ_{1}, λ_{2} \in \partial^{-} {\tilde{Ω}}_{1} .

Let

l_{0}

be the segment

[λ, λ_{1}]

in

Λ,

and let

l_{1}

be the image of

{\tilde{Ω}}_{1}

in

Λ .

By the above construction,

s_{ξ}

is already defined over

l_{0},

λ_{1}

and

l_{1} .

It is a consequence of Lemma 4.3 and Theorem 4.4 that

{s_{ξ} |}_{l_{1}}

has a well-defined endpoint at

\partial_{\infty} λ_{2}

(see [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

, Lemma 6.18]). We extend

s_{ξ}

to

λ_{2}

continuously, and follow the rightmost-down rule to define it over the segment

l_{2} := [λ_{2}, λ_{3}] .

We continue along the sequence

λ_{i}

until we have defined

s_{ξ} (μ) .

Since the dual graph

Λ^{*}

is a tree, there is a unique way to reach any incomparable

μ

from

λ

through such a sequence of

λ_{n} .

In the end, we have a section

s_{ξ}

that is well defined on the whole

Λ .

The process of extending

s_{ξ}

is a process of branching out from

λ

and sweeping

Λ .

The values at leaves that are closer to

λ

are defined first, and the values at leaves farther away from

λ

are determined by the closer values. At each point of

Λ,

there is a direction of extension of

s_{ξ}

that points toward the direction away from

λ,

along which

s_{ξ}

is defined.

Figure 12.

Extending a leftmost section to incomparable leaves.

There is a unique leftmost section starting from any point in

E_{\infty} .

The set of leftmost sections is denoted by

LS .

The images of two different leftmost sections might coalesce but can never cross each other. If

ℓ

is a line in

Λ,

the bundle

{E_{\infty} |}_{ℓ}

is homeomorphic to a cylinder, and the leftmost sections restricted to

ℓ

give embedded lines on

{E_{\infty} |}_{ℓ}

transverse to the fiber. For any three different leftmost sections, there is an embedded line in

Λ

so that the restrictions of the sections to this line have a well-defined cyclic order, and the cyclic order is independent of the choice of the line [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

, Lemma 6.25]. The completion of

LS

with respect to the cyclic order is homeomorphic to a circle, denoted by

𝔖_{left} .

The fundamental group

π_{1} (M)

acts naturally on

LS,

and the action extends to an action on

𝔖_{left} .

For any

λ \in Λ,

there is a valuation map

U_{λ} : LS \to \partial_{\infty} λ

given by

U_{λ} (s) = s (λ) .

The map

U_{λ}

can be extended to a monotone map

U_{λ} : 𝔖_{left} \to \partial_{\infty} λ .

Theorem 4.5 ([7
D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363
]).

The circle $𝔖_{left}$ together with the $π_{1} (M) -$ action and the set of structure maps ${U_{λ}}_{λ \in Λ}$ is a universal circle for $ℱ .$

In order to prove Theorem 1.2, we need to analyze more carefully what marker ends and the leftmost sections look like on

E_{\infty} .

We will continue to use the terminologies from Section 3.

4.2. Markers contained in a product region

We first consider the ends of markers that are contained in a product region. The identification of

\tilde{Ω}

with

\tilde{L} \times ℝ

gives a canonical identification of

{E_{\infty} |}_{\tilde{Ω}}

as

\partial_{\infty} \tilde{L} \times ℝ .

Here we implicitly use that for any homeomorphism between two infinite-type surfaces with standard hyperbolic structures, any lift to their universal covers extends continuously to a homeomorphism between their boundaries at infinity, and the extension is unique [10

J. Cantwell and L. Conlon, Hyperbolic geometry and homotopic homeomorphisms of surfaces. Geom. Dedicata 177 (2015), no. 1, 27–42 Zbl 1359.37097 MR 3370020

]. Denote the leaf

\tilde{L} \times {t}

by

λ_{t} .

Again, each

λ_{t}

is identified with

\tilde{L} .

Lemma 4.6.

For any $ξ \in \partial_{\infty} λ_{t},$ there is an $ε > 0$ so that ${ξ} \times [t, t + ε]$ is the end of a marker.

Proof.

We will use a tightening method described in [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

, Section 5.3]. Take any point

x \in λ_{t}

and consider the geodesic ray

γ

from

x

to

ξ .

Since depth-one leaves in the same fibered region have asymptotic ends, for every small

δ > 0,

there is an

ε

such that any flowline of

\tilde{ϕ}

between

λ_{t}

and

λ_{t + ε}

has length

< δ .

Moreover, the map from

λ_{t}

to any

λ_{s}

with

s \in (t, t + ε]

induced by flowing

λ_{t}

to

λ_{s}

is

K -

bi-Lipschitz for some uniform

K .

Therefore, the flow image

γ_{s}

of

γ

in each such

λ_{s}

is a family of uniform quasi-geodesics with ideal endpoints

(ξ, s) .

We can then tighten

γ_{s}

to geodesics

γ_{s}^{*}

on

λ_{s} .

We claim that the union of the

λ_{s}^{*}

for

s \in [t, t + ε]

is a continuous one-ended band with bounded width. That is because any pair of

γ_{s_{1}}

and

γ_{s_{2}}

are bounded Hausdorff distance from each other, and so are their geodesic tightenings

γ_{s_{1}}^{*}

and

γ_{s_{2}}^{*} .

By the continuity of the leafwise hyperbolic metric, this only happens when

γ_{s}^{*}

is a continuous family of geodesic rays. The union of

γ_{s}^{*}

has bounded width because the tightening process only shifts the rays by a bounded amount. Finally, we can take

ε

even smaller to obtain a genuine marker with width

< ε_{0}

and with the end

{ξ} \times [t, t + ε] .

The next corollary follows immediately from the construction of leftmost sections on comparable leaves.

Corollary 4.7.

(Leftmost sections on a product region are vertical) Suppose $s$ is any leftmost section of $E_{\infty} .$ If $s (λ_{t_{0}}) = (ξ, t_{0})$ for some $t_{0} \in ℝ,$ then $s (λ_{t}) = (ξ, t)$ for all $t .$

Proof.

By the construction of leftmost sections, one can see that any leftmost section

s

has the following property: when extending

s

upward, if

s

meets the end

e

of a marker at a point

x \in e,

then

s

has to contain the part of

e

above

x .

The same is true if we are extending

s

downward: if

s

meets

e

at a point

x,

it must also contain everything in

e

below

x .

Since through any point in

{E_{\infty} |}_{\tilde{Ω}}

there is a vertical marker end in both directions, the leftmost section

s

restricted to

\tilde{Ω}

is forced to be vertical.

4.3. Markers intersecting a type-0 leaf

We now consider the ends of markers intersecting a type-0 leaf. Let

μ

be a type-0 leaf covering a compact leaf

Σ,

and assume for the rest of this section that

μ

is in the positive boundary of

\tilde{Ω} .

The case where

μ

is in the negative boundary of

\tilde{Ω}

is similar.

As in the previous discussion, we identify every depth-one leaf in

Ω

with

L,

and every type-1 leaf in

\tilde{Ω}

with

\tilde{L} .

This gives us a homeomorphism between

{E_{\infty} |}_{\tilde{Ω}}

and

\partial_{\infty} \tilde{L} \times ℝ .

Recall that in Section 3, we define a continuous map

I_{λ_{t} μ} : \partial_{\infty} λ_{t} \to \partial_{\infty} μ

for any

λ_{t} \in \tilde{Ω} .

Under the homeomorphism

\partial_{\infty} λ_{t} ≅ \partial_{\infty} \tilde{L},

the map

I_{λ_{t} μ}

is the same map for any

t

when viewed as a map from

\partial_{\infty} \tilde{L}

to

\partial_{\infty} μ .

We denote this map by

I_{\tilde{Ω} μ} .

In

Λ,

\tilde{Ω} \cup μ

is a half-open interval. Let

ℳ_{μ, \tilde{Ω}}

be the set of markers

m

in

μ \cup \tilde{Ω}

with one side lying in

μ .

By Corollary 4.7 and Lemma 4.3, the end of such an

m

intersects

\partial_{\infty} μ

at a single point

ξ_{μ} (m)

and intersects

{E_{\infty} |}_{\tilde{Ω}}

in a vertical segment

{(ξ_{\tilde{Ω}} (m), t) ∣ t > T},

where

T

is a constant depending on

m,

and

ξ_{\tilde{Ω}} (m)

is a point in

\partial_{\infty} \tilde{L}

depending on

m

and independent of

t

(Figure 13).

Figure 13.

The end of a marker

m

(in red) in

ℳ_{μ, \tilde{Ω}} .

Define the set

ξ_{\tilde{Ω}} (ℳ_{μ, \tilde{Ω}}) := {ξ_{\tilde{Ω}} (m) ∣ m \in ℳ_{μ, \tilde{Ω}}} \subset \partial_{\infty} \tilde{L} .

Intuitively, these are the directions on

\tilde{L}

in which

μ

does not diverge from

λ_{t} .

Recall that the positive escaping set

𝒰^{+} \subset L

is the set of points whose

ϕ -

orbit escapes

Ω

in positive time. Let

{\tilde{𝒰}}^{+}

be the preimage of

𝒰^{+}

in

\tilde{L} .

There is a component

{\tilde{𝒰}}_{μ}^{+}

of

{\tilde{𝒰}}^{+}

so that

x \in λ_{t}

is a point of

{\tilde{𝒰}}_{μ}^{+}

if and only if

\tilde{ϕ} (x)

hits

μ .

The hitting map

H_{μ} : {\tilde{𝒰}}_{μ}^{+} \to μ

defined by

H_{μ} (x) = \tilde{ϕ} (x) \cap μ

is a homeomorphism since it is an open bijection. It is tautological that

(4.1)

{p |}_{μ} \circ H_{μ} = {p |}_{{\tilde{𝒰}}_{μ}^{+}},

where

p

is the projection to

𝒪

as in Section 3.

Lemma 4.8.

Let $α^{*}$ be an oriented simple closed geodesic in $Σ$ with trivial $ℱ -$ holonomy on the side of $Ω,$ and let ${\tilde{α}}^{*}$ be a lift of $α^{*}$ to $μ .$ Denote the forward endpoint at infinity of ${\tilde{α}}^{*}$ by $\partial^{+} {\tilde{α}}^{*} .$ Then there is a marker $m \in ℳ_{μ, \tilde{Ω}}$ so that $ξ_{μ} (m) = \partial^{+} {\tilde{α}}^{*}$ and $ξ_{\tilde{Ω}} (m) \in I_{\tilde{Ω} μ}^{- 1} (\partial^{+} {\tilde{α}}^{*}) .$

Proof.

Perform a homotopy of

α^{*}

along

ϕ

into

Ω

for a short distance so that the final image is a simple closed curve on a depth-one leaf in

Ω .

This is possible because

α^{*}

has trivial

ℱ -

holonomy on the side of

Ω .

The full homotopy image is an annulus, denoted by

A .

We lift

A

to

\tilde{M}

to get a two-ended infinite band

\tilde{A}

in

\tilde{Ω} \cup μ

with one side being

{\tilde{α}}^{*}

and the other side on

λ_{T}

for some

T .

Note that

A

has bounded width because it is a lift of an annulus.

Fix a base point

x \in {\tilde{α}}^{*},

and let

{\tilde{α}}^{*} (x)

be the oriented ray in

{\tilde{α}}^{*}

starting from

x

toward

\partial^{+} {\tilde{α}}^{*} .

We restrict the lifted homotopy to the ray

{\tilde{α}}^{*} (x),

and the restricted homotopy image is a one-ended infinite band

B \subset \tilde{A}

(Figure 14). The band

B

has finite width between

μ

and

λ_{T},

and we can make the width arbitrarily small by cutting the homotopy at

λ_{T^{'}}

for large enough

T^{'} .

When

t > T^{'},

the ray

B \cap λ_{t}

has a well-defined endpoint at infinity because it is contained in a lift of a simple closed curve on a hyperbolic depth-one leaf. Similar to the proof of Lemma 4.6 and the tightening method in [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

, Section 5.3], by taking even larger

T^{'}

and pulling tight the intersection of

B

and

λ_{t},

we get a marker

m \in ℳ_{μ, \tilde{Ω}} .

Note that different from Lemma 4.6, here we do not have uniform bi-Lipschitz maps from

μ

to type-1 leaves in

\tilde{Ω} .

However, we still have that the rays

B \cap λ_{t}

are a family of uniform quasi-geodesics by the continuity of the leafwise metric, so the same tightening argument as in [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

, Section 5.3] still works. It is clear from the construction that

ξ_{μ} (m) = \partial^{+} {\tilde{α}}^{*} .

The point

ξ_{\tilde{Ω}} (m)

is the endpoint of the ray

B \cap λ_{t} = H_{μ}^{- 1} ({\tilde{α}}^{*} (x)) .

By (4.1), the projections of

{\tilde{α}}^{*} (x)

and

H_{μ}^{- 1} ({\tilde{α}}^{*} (x))

to

𝒪

are identical, and both rays escape to infinity in the leaf. This implies

\partial^{+} {\tilde{α}}^{*} \in \partial p (μ) \cap \partial p (λ_{t})

and

ξ_{\tilde{Ω}} (m) \in I_{\tilde{Ω} μ}^{- 1} (\partial^{+} {\tilde{α}}^{*}) .

Figure 14.

An annulus without holonomy (shaded on the left) lifts to a band

B

(shaded on the right) that gives a marker.

To state the next lemma, we need one more definition. A simple closed curve

β \subset L

is called a $Σ -$ juncture if

β

is a connected positive

f -

juncture and

β

covers a simple closed curve in

Σ

under the covering map

𝒰^{+} \to \partial^{+} Ω

(note that

Σ

is only a component of

\partial^{+} Ω

).

Corollary 4.9.

Let $β \subset L$ be a $Σ -$ juncture, and let $\tilde{β}$ be a lift of $β$ in $\tilde{L}$ that lies in ${\tilde{𝒰}}_{μ}^{+} .$ Then both endpoints of $\tilde{β}$ are in $ξ_{\tilde{Ω}} (ℳ_{μ, \tilde{Ω}}) .$

Proof.

Suppose

β

covers a simple closed curve

α

on

Σ .

Fix an orientation of

β,

which induces an orientation of

α .

The curve

α

has trivial

ℱ -

holonomy on the side of

Ω

because

β

is an

f -

juncture. For any

T,

we can view

\tilde{β}

as an embedded line in

λ_{T} .

There is a lift

\tilde{α} \subset μ

of

α

given by

\tilde{α} = H_{μ} (\tilde{β}) .

Let

\partial^{+} \tilde{α}

be the forward endpoint of

\tilde{α} .

Applying Lemma 4.8 to the geodesic tightening

α^{*}

of

α,

we obtain a marker

m \in ℳ_{μ, \tilde{Ω}} .

But since

\tilde{α}

is obtained by flowing

\tilde{β}

to

μ,

the proof of Lemma 4.8 implies that

ξ_{\tilde{Ω}} (m)

is exactly the forward endpoint of

\tilde{β} .

The same proof applies for the backward endpoint of

\tilde{β},

proving the corollary.

We define the limit set

ℒ ({\tilde{𝒰}}_{μ}^{+})

of

{\tilde{𝒰}}_{μ}^{+}

as the intersection

cl ({\tilde{𝒰}}_{μ}^{+}) \cap \partial_{\infty} \tilde{L},

where

cl ({\tilde{𝒰}}_{μ}^{+})

is defined to be the closure taken in

\tilde{L} \cup \partial_{\infty} \tilde{L} ≅ \bar{ℍ^{3}} .

Lemma 4.10.

The closure in $\partial_{\infty} \tilde{L}$ of $ξ_{\tilde{Ω}} (ℳ_{μ, \tilde{Ω}})$ is $ℒ ({\tilde{𝒰}}_{μ}^{+}) .$

Proof.

Suppose

m

is a marker in

ℳ_{μ, \tilde{Ω}} .

The intersection of the image of

m

with any

λ_{t}

is a ray

r_{t} .

After identifying

λ_{t}

with

L,

we claim that

r_{t}

is contained in

{\tilde{𝒰}}_{μ}^{+} .

This is because the image of

m

is

ε_{0} -

close to

μ .

By our choice of

ε_{0},

the

ε_{0} -

neighborhood of

Σ

in

M

is contained in an

ℱ -

spiraling neighborhood

N (Σ)

of

Σ,

and every

ϕ -

orbit intersecting the spiraling neighborhood will hit

Σ .

So

r_{t}

is contained in

{\tilde{𝒰}}_{μ}^{+},

and the ideal endpoint of

r_{t},

which is exactly

ξ_{\tilde{Ω}} (m)

by definition, will then be contained in

ℒ ({\tilde{𝒰}}_{μ}^{+}) .

This shows

ξ_{\tilde{Ω}} (ℳ_{μ, \tilde{Ω}}) \subset ℒ ({\tilde{𝒰}}_{μ}^{+}) .

By Corollary 4.9, it now suffices to show that the endpoints of lifts of any

Σ -

juncture in

{\tilde{𝒰}}_{μ}^{+}

are dense inside

ℒ ({\tilde{𝒰}}_{μ}^{+}) .

We first recall some known facts and constructions. By [11

J. Cantwell, L. Conlon, and S. R. Fenley, Endperiodic automorphisms of surfaces and foliations. Ergodic Theory Dynam. Systems 41 (2021), no. 1, 66–212 Zbl 1464.57022 MR 4190052

], the geodesic tightenings of a system of positive

f -

junctures limit to the negative Handel–Miller geodesic lamination

Λ_{HM}^{-}

under negative iterations of

f,

and

Λ_{HM}^{-}

is independent of the choice of junctures. On the other hand, the intersection of

ℱ^{s}

with

L

induces a singular foliation

ℱ_{L}^{s}

on

L .

We define

W^{-}

as the restriction of

ℱ_{L}^{s}

to the complement of

𝒰^{+} .

The complement of

𝒰^{+}

is saturated by leaves of

ℱ_{L}^{s}

and

W^{-}

is a singular sublamination of

ℱ_{L}^{s}

by [23

M. P. Landry, Y. N. Minsky, and S. J. Taylor, Endperiodic maps via pseudo-Anosov flows. 2023, arXiv:2304.10620v1, to appear in Geom. Topol.

]. Let

{\tilde{W}}^{-}

be the lift of

W^{-}

to

\tilde{L} .

The singular lamination

{\tilde{W}}^{-}

determines an abstract lamination on

\partial_{\infty} \tilde{L}

by a standard construction (Section 2). It follows from [23

M. P. Landry, Y. N. Minsky, and S. J. Taylor, Endperiodic maps via pseudo-Anosov flows. 2023, arXiv:2304.10620v1, to appear in Geom. Topol.

, Theorem 8.4] that

{\tilde{W}}^{-} = Λ_{HM}^{-}

as abstract laminations (the paper proves it for circular pseudo-Anosov flows, but the same method applies to any pseudo-Anosov flow without perfect fits).

Now fix a

Σ -

juncture

β .

Suppose that

β

is the boundary of a contracting neighborhood of a contracting end

ℰ,

and

n

is the smallest positive integer so that

f^{n} (ℰ) = ℰ .

The above facts imply that the endpoints of

\partial {\tilde{𝒰}}_{μ}^{+}

can be approximated by endpoints of lifts of

{f^{- i n} (β)}_{i \geq 0} .

The limit set

ℒ ({\tilde{𝒰}}_{μ}^{+})

is nowhere dense by the next lemma (Lemma 4.11), so points in

ℒ ({\tilde{𝒰}}_{μ}^{+})

are approximated by endpoints of

\partial ℒ ({\tilde{𝒰}}_{μ}^{+}) .

Also note that if

β

is a

Σ -

juncture, so is

f^{- i n} (β) .

Therefore, the endpoints of lifts of any

Σ -

juncture in

{\tilde{𝒰}}_{μ}^{+}

are dense inside

ℒ ({\tilde{𝒰}}_{μ}^{+}),

which completes the proof.

We give a proof of the following fact used in the proof of Lemma 4.10.

Lemma 4.11.

The limit set $ℒ ({\tilde{𝒰}}_{μ}^{+})$ is nowhere dense in $\partial_{\infty} \tilde{L} .$

Proof.

The subsurface

𝒰^{+} \subset L

is bounded by leaves of

ℱ_{L}^{s}

[23

M. P. Landry, Y. N. Minsky, and S. J. Taylor, Endperiodic maps via pseudo-Anosov flows. 2023, arXiv:2304.10620v1, to appear in Geom. Topol.

], each of which has distinct well-defined endpoints at infinity. In particular,

ℒ ({\tilde{𝒰}}_{μ}^{+})

is a proper subset of

\partial_{\infty} \tilde{L} .

If

ℒ ({\tilde{𝒰}}_{μ}^{+})

contains a non-trivial interval

I,

we can find a hyperbolic element

g \in π_{1} (L)

with a fixed point in

I .

Here we are using the fact that

L

has bounded injectivity radius. But

g ({\tilde{𝒰}}_{μ}^{+})

is either disjoint from or equal to

{\tilde{𝒰}}_{μ}^{+}

since

𝒰^{+}

is embedded. So

g^{n} (I)

is contained in

ℒ ({\tilde{𝒰}}_{μ}^{+})

for any integer

n,

a contradiction.

Lemma 4.12.

There is a subset $ℳ_{μ, \tilde{Ω}}^{'}$ of $ℳ_{μ, \tilde{Ω}}$ such that

the set $ξ_{μ} (ℳ_{μ, \tilde{Ω}}^{'})$ is dense in $\partial_{\infty} μ$ ;
the set $ξ_{\tilde{Ω}} (ℳ_{μ, \tilde{Ω}}^{'})$ is dense in $ℒ ({\tilde{𝒰}}_{μ}^{+})$ ;
for any $m \in ℳ_{μ, \tilde{Ω}}^{'}$ , we have

$ξ_{μ} (m) = I_{\tilde{Ω} μ} (ξ_{\tilde{Ω}} (m)) .$

Proof.

The subset

ℳ_{μ, \tilde{Ω}}^{'}

can be taken to be the set of markers in

ℳ_{μ, \tilde{Ω}}

that arise from Lemma 4.8. Item (3) is exactly the content of Lemma 4.8. Item (1) is true because the endpoints of the lifts of a simple closed curve are dense in

\partial_{\infty} μ .

The second part of the proof of Lemma 4.10 only used markers in

ℳ_{μ, \tilde{Ω}}^{'},

so we have already proved item (2).

Lemma 4.13.

If $η$ is a point in $\partial_{\infty} μ$ such that $I_{\tilde{Ω} μ}^{- 1} (η)$ is a closed interval with positive length, then $η \notin ξ_{μ} (ℳ_{μ, \tilde{Ω}}) .$

Proof.

Suppose there is a marker

m \in ℳ_{μ, \tilde{Ω}}

such that

ξ_{μ} (m) = η .

Let

r_{μ}

be the geodesic ray

m \cap μ,

and let

r_{t}

be the geodesic ray

m \cap λ_{t} .

Let

e

be the side of

p (μ)

corresponding to

η .

Then the interior of

e

is contained in

p (λ_{t})

for all

t .

Take a quasi-geodesic ray

r^{'}

in

μ

such that

p (r^{'})

has an endpoint in the interior of

e .

For example, using the notations in Proposition 3.1, for any

x \in e,

we can take

r^{'}

to be a ray in

l_{x}

with one end at

x .

Since

r^{'}

and

r_{μ}

both have endpoint

μ,

they have bounded Hausdorff distance from each other on

μ .

By the choice of

ε_{0}

in Definition 4.2, each

r_{t}

in

m

flows forward in bounded time to hit

μ,

and the image on

μ

is exactly

H_{μ} (r_{t}) .

Therefore,

r^{'}

has bounded Hausdorff distance from

r_{t}

and

H_{μ} (r_{t})

as well. Since

ε_{0}

is smaller than a separation constant (Definition 4.2),

r^{'}

and

H_{μ} (r_{t})

are also a bounded Hausdorff distance from each other in

μ .

Now fix a

λ_{t}

intersecting

m,

and for any

y \in μ,

define

T (y)

to be the time it takes for

y

to flow backward to

λ_{t} .

If we write

{\tilde{ϕ}}^{T}

for flowing along

\tilde{ϕ}

for time

T,

then

T (y)

satisfies

ϕ^{- T (y)} (y) \in λ_{t} .

This function

T

is bounded on

H_{μ} (r_{t})

by the above. On the other hand, since we pick

r^{'}

to have an endpoint in the interior of

p (λ_{t}),

T (y)

must go to

+ \infty

as

y

travels along

r^{'}

to infinity. We will show that this is a contradiction. To see this, take sequences

y_{n} \in H_{μ} (r_{t})

and

z_{n} \in r^{'}

with

d_{μ} (y_{n}, z_{n}) \leq C,

where

C

is a constant, and with

x_{n}

(and hence

y_{n}

) going to infinity. Then

T (y_{n})

is bounded, while

T (z_{n})

goes to

+ \infty .

By acting with deck transformations in

π_{1} (μ),

we can find translates

(y_{n}^{'}, z_{n}^{'})

of

(y_{n}, z_{n})

that stay in a compact subset of

μ .

Since the transformations preserve

\tilde{ℱ},

{\tilde{ϕ}}^{- T (y_{n})} (y_{n}^{'})

and

{\tilde{ϕ}}^{- T (z_{n})} (z_{n}^{'})

are in the same type-1 leaf in

\tilde{Ω} .

Assume

y_{n}^{'}

converges to

y

and

z_{n}^{'}

converges to

z,

and up to taking a subsequence, assume that

T (y_{n})

converges to a finite positive number

T_{0} .

Then the flowline

\tilde{ϕ} (z)

will never intersect the type-1 leaf containing

{\tilde{ϕ}}^{- T_{0}} (y),

which is a contradiction because every flowline intersecting

\tilde{Ω}

will intersect every leaf in

\tilde{Ω} .

This proves that such a marker

m

does not exist.

The following lemmas, Lemmas 4.14 and 4.15, together with Corollary 4.7, will give us the complete rules to build the leftmost section starting from a given point. Lemma 4.14 tells us how to extend the leftmost section from a product region to an adjacent type-0 leaf, and Lemma 4.15 tells us how to go from a type-0 leaf to an adjacent product region.

For any

ξ \in \partial_{\infty} \tilde{L},

we use

c_{ξ}

to denote the vertical section of

{E_{\infty} |}_{\tilde{Ω}}

given by

c_{ξ} (λ_{t}) = (ξ, t) .

Lemma 4.14.

For any $ξ \in \partial_{\infty} (\tilde{L}),$ the vertical section $c_{ξ}$ extends continuously to $μ$ by setting $c_{ξ} (μ) = I_{\tilde{Ω} μ} (ξ) .$

Proof.

First of all, we remark that as noted in [7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

, Proof of Lemma 6.18], the closure of

c_{ξ}

in

E_{\infty}

is a closed interval transverse to the circle fibers, intersecting

\partial_{\infty} μ

in exactly one point. This is a consequence of the density of markers, and it implies the section

c_{ξ}

has a unique continuous extension to

μ .

To determine the value of

c_{ξ}

at

μ,

we first consider the case where

ξ

is in a gap

G_{ξ}

of

I_{\tilde{Ω} μ} .

Let

\partial_{r} G_{ξ}

and

\partial_{l} G_{ξ}

be the leftmost and the rightmost endpoints of

G_{ξ},

respectively. By Lemma 4.12, we have sequences

{m_{n}^{+}}

and

{m_{n}^{-}}

in

ℳ_{μ, \tilde{Ω}}^{'} \subset ℳ_{μ, \tilde{Ω}}

with the following properties:

the points $ξ_{\tilde{Ω}} (m_{n}^{-})$ limit to $\partial_{l} G_{ξ}$ from the left and the points $ξ_{\tilde{Ω}} (m_{n}^{+})$ limit to $\partial_{r} G_{ξ}$ from the right;
we have $ξ_{μ} (m_{n}^{\pm}) = I_{\tilde{Ω} μ} (ξ_{\tilde{Ω}} (m_{n}^{\pm})) .$

These properties imply that the points

ξ_{μ} (m_{n}^{-})

limit to

I_{\tilde{Ω} μ} (ξ)

from the left and the points

ξ_{μ} (m_{n}^{+})

limit to

I_{\tilde{Ω} μ} (ξ)

from the right by item (3) of Lemma 4.12 and Lemma 4.3. The two sequences of marker ends pin down the endpoint of

c_{ξ}

to be

I_{\tilde{Ω} μ} (ξ)

(see the left-hand side of Figure 15).

Figure 15.

The subset

ℳ_{μ, \tilde{Ω}}^{'}

(red) pins down the way to extend the leftmost sections (blue), with the blue arrows indicating the direction of extension. The left-hand side corresponds to Lemma 4.14 and the right-hand side corresponds to Lemma 4.15.

When

ξ

is not in any gap of

I_{\tilde{Ω} μ},

the proof can be done similarly to above by replacing

G_{ξ},

\partial_{r} G_{ξ}

and

\partial_{l} G_{ξ}

all by the point

ξ .

Lemma 4.15.

Let $η$ be a point in $\partial_{\infty} μ .$ Suppose $ν$ is any leaf of $\tilde{ℱ},$ $ξ$ is any point in $\partial_{\infty} ν$ and $s_{ξ}$ is the leftmost section starting from $ξ .$ If the direction of extension of $s_{ξ}$ at $μ$ points from $μ$ to $\tilde{Ω},$ that is, $μ$ is closer to $ν$ in $Λ$ than $\tilde{Ω},$ and $s_{ξ} (μ) = η,$ then $s_{ξ} (λ_{t})$ is the rightmost endpoint of $I_{\tilde{Ω} μ}^{- 1} (η) .$ Here we view a singleton as a closed interval of length zero.

Remark 4.16.

We remind the readers that here we are assuming

μ

to be in the positive boundary of

\tilde{Ω},

as stated in the first paragraph of Section 4.3, so the direction of extension of

s_{ξ}

is the backward flow direction. If

μ

is in the negative boundary of

\tilde{Ω}

and the direction of extension of

s_{ξ}

is the forward flow direction pointing from

μ

to

\tilde{Ω},

the lemma remains true if we replace “the rightmost endpoint” by “the leftmost endpoint” with the same proof.

Proof.

We refer the readers to Figure 15 for an illustration of the situation. The value of

s_{ξ}

on

\tilde{Ω}

is determined by going down from

η

and following the rightmost-down rule, described in Section 4.1. Since, by Corollary 4.7,

s_{ξ}

can only go vertically down in

\tilde{Ω},

its value in

\tilde{Ω}

is the leftmost infimum of

ξ_{\tilde{Ω}} (m)

over all

m \in ℳ_{μ, \tilde{Ω}}

such that

ξ_{μ} (m)

is in a small neighborhood

U

of

η

and

ξ_{μ} (m)

is not on the left of

η

(i.e., after identifying

U

with an interval in

ℝ,

η \leq ξ_{μ} (m)

). By Lemma 4.12, there is a sequence of markers

m_{n} \in ℳ_{μ, \tilde{Ω}}^{'} \subset ℳ_{μ, \tilde{Ω}}

so that

ξ_{μ} (m_{n})

limits to

η

from the right. By the monotonicity of

I_{\tilde{Ω} μ}

and item (3) of Lemma 4.12, we have that

ξ_{\tilde{Ω}} (m_{n})

limits to the rightmost endpoint of

I_{\tilde{Ω} μ}^{- 1} (η)

from the right. If

I_{\tilde{Ω} μ}^{- 1} (η)

is a single point, this implies that

s_{ξ} (λ_{t}) = I_{\tilde{Ω} μ}^{- 1} (η) .

If

I_{\tilde{Ω} μ}^{- 1} (η)

is a closed interval, the existence of

m_{n}

and Lemma 4.13 imply

s_{ξ} (λ_{t})

is the rightmost endpoint of

I_{\tilde{Ω} μ}^{- 1} (η) .

5. Building the homeomorphism

Suppose

λ

is a leaf of

ℱ

and

ξ

is a point in

\partial_{\infty} λ .

Let

V_{ξ}

denote

I_{λ}^{- 1} (ξ) \subset \partial 𝒪,

which is either a closed interval or a singleton. If

V_{ξ}

is a closed interval, let

\partial_{l} V_{ξ}

and

\partial_{r} V_{ξ}

be the leftmost and the rightmost endpoints of

V_{ξ},

respectively. Note that by our convention, left means clockwise and right means counterclockwise. We say

V_{ξ}

is a stable (resp. unstable) gap of

λ

if

Q_{λ}^{- 1} (ξ)

is a boundary

ℱ_{𝒪}^{s} -

leaf (resp.

ℱ_{𝒪}^{u} -

leaf) of

p (λ) .

Note that a closed interval of

\partial 𝒪

cannot be a stable gap of a leaf while being an unstable gap of another. Indeed, a leaf of

ℱ_{𝒪}^{s}

and a leaf of

ℱ_{𝒪}^{u}

cannot bound an ideal bigon. In fact, for

ϕ

without perfect fits, a stronger statement is true: any two rays in

ℱ_{𝒪}^{s} \cup ℱ_{𝒪}^{u}

have distinct endpoints at infinity [13

S. Fenley, Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry. Geom. Topol. 16 (2012), no. 1, 1–110 Zbl 1279.37026 MR 2872578

, Lemma 3.20]. We say a closed interval of

\partial 𝒪

is a stable (resp. an unstable) gap if it is a stable (resp. an unstable) gap of a leaf of

\tilde{ℱ} .

Since

𝔖_{left}

is the completion of the set

LS

of leftmost sections, to define the homeomorphism

T : 𝔖_{left} \to \partial 𝒪,

it suffices to define

T

on

LS

and show that it admits an extension. For any point

ξ

in

E_{\infty},

let

s_{ξ}

be the leftmost section starting from

ξ .

The set

{LS}^{*}

of pointed leftmost sections is the set

{(s, ξ) ∣ s \in LS, ξ \in E_{\infty}, s = s_{ξ}} .

We point out that this definition is not redundant, for it is possible to have

s_{ξ} = s_{ξ}^{'}

with

ξ \neq ξ^{'} .

There is a natural forgetful map

π : {LS}^{*} \to LS

given by forgetting the starting point. We define a map

T^{*} : {LS}^{*} \to \partial 𝒪

as follows. For a pointed leftmost section

(s, ξ),

consider

V_{ξ} .

If

V_{ξ}

is a single point, define

T^{*} (s, ξ) = V_{ξ} .

If

V_{ξ}

is an unstable gap, define

T^{*} (s, ξ) = \partial_{l} V_{ξ} .

If

V_{ξ}

is a stable gap, define

T^{*} (s, ξ) = \partial_{r} V_{ξ} .

Theorem 1.2 follows from the following theorem.

Theorem 5.1.

The map $T^{*}$ descends to a map $T : LS \to \partial 𝒪$ that extends continuously to a map from $𝔖_{left}$ to $\partial 𝒪,$ which we will again denote by $T .$ The extension $T$ is injective and preserves the cyclic order; hence it is a homeomorphism. Moreover, $T$ is $π_{1} (M) -$ equivariant, and for any $λ \in Λ$ and any $s \in 𝔖_{left},$ we have $U_{λ} (s) = I_{λ} (T (s)) .$ In other words, $T$ is an isomorphism of universal circles between $𝔖_{left}$ and $\partial 𝒪 .$

The rest of this section will be dedicated to proving Theorem 5.1.

Lemma 5.2.

Suppose $(s, ξ_{1})$ is an element in ${LS}^{*} .$ Let $μ$ be any leaf of $\tilde{ℱ}$ and suppose $s (μ) = ξ .$ Then $T^{*} (s, ξ_{1}) \in V_{ξ} .$

Proof.

By definition, we have

s = s_{ξ_{1}}

is the leftmost section starting from

ξ_{1},

where

ξ_{1} \in \partial_{\infty} λ_{1}

for some

λ_{1} \in Λ .

We assume that

λ_{1}

is a type-1 leaf for simplicity. The case when

λ_{1}

is type-0 is basically the same.

We first consider the case when

λ_{1}

and

μ

are comparable and

λ_{1} < μ .

Take a sequence of

\tilde{ℱ} -

leaves

λ_{1} \to λ_{2} \to λ_{3} \to \dots \to λ_{n} = μ,

where

λ_{2 i}

is a type-0 leaf,

λ_{2 i + 1}

is a type-1 leaf (note that here we set up the notations so that

λ_{k}

is of type

k

mod 2) and

λ_{i} ≲ λ_{i + 1} .

The sequence represents the shortest path from

\tilde{Ω} {(λ_{1})}^{*}

to

μ^{*}

in

Λ^{*},

after identifying a type-0 leaf with the dual vertex and a type-1 leaf with the vertex dual to the product region containing it. We record the value of

s

along this sequence by

s_{i} = s (λ_{i}),

and let

V_{i} = V_{s (λ_{i})} .

We have a sequence of closed intervals (possibly with length zero)

V_{i} \subset \partial 𝒪 .

The goal is to show that for all

i = 1, \dots, n,

we have

T^{*} (s, ξ_{0}) \in V_{i} .

In particular, this implies

T^{*} (s, ξ_{0}) \in V_{n} = V_{ξ} .

We will show this by tracking how the intervals

V_{i}

vary along the sequence

λ_{i} .

By Lemma 4.14, we have

V_{2 i - 1} \subset V_{2 i};

by Lemma 4.15, we have

V_{2 i + 1} \subset V_{2 i} .

Lemma 5.3.

If

V_{2 i - 1} ⊊ V_{2 i},

then

V_{2 i}

is a stable gap. If

V_{2 i + 1} ⊊ V_{2 i},

then

V_{2 i}

is an unstable gap and

V_{2 i + 1} = \partial_{l} V_{2 i} .

Proof.

First, suppose

V_{2 i - 1} ⊊ V_{2 i} .

The leaf

e := Q_{λ_{2 i}}^{- 1} (I_{λ_{2 i}} (V_{2 i}))

is a side of

p (λ_{2 i})

containing in the interior of

p (λ_{2 i - 1}) .

The fixed point

x_{e}

in

e

under

Stab (e)

corresponds to a periodic

\tilde{ϕ} -

orbit

γ

in

\tilde{M}

intersecting

λ_{2 i - 1}

but not intersecting

λ_{2 i} .

Since every

\tilde{ϕ} -

orbit in

{\tilde{ℱ}}^{s} (ϕ)

is forward asymptotic to

γ,

we see that

ℱ_{𝒪}^{s} (x)

is not contained in

p (λ_{2 i}) .

This means

e = ℱ_{𝒪}^{s} (x)

and

V_{2 i}

is a stable gap.

Now suppose

V_{2 i + 1} ⊊ V_{2 i} .

A similar argument to the above shows

V_{2 i}

is an unstable gap. By Lemma 4.15 and Remark 4.16, if

V_{2 i + 1}

is a closed interval of positive length, there will be a boundary leaf

e

of

p (λ_{2 i})

and a boundary leaf

e^{'}

of

p (λ_{2 i + 1})

different from

e

and sharing the leftmost endpoint with

e .

But this cannot happen because when

ϕ

has no perfect fits, no pair of leaves in

ℱ_{𝒪}^{s} \cup ℱ_{𝒪}^{u}

can share an endpoint. Therefore,

V_{2 i + 1}

is a single point, and it is the leftmost endpoint of

V_{2 i}

by Lemma 4.15.

We continue the proof that

T^{*} (s, ξ_{1}) \in V_{i}

for all

1 \leq i \leq n .

It is obvious that

T^{*} (s, ξ_{1}) \in V_{1}

by the definition of

T^{*} .

If

V_{1}

is a single point or a stable gap of

λ_{1},

by Lemma 5.3 we have

V_{i} \subset V_{i + 1}

for all

i .

So we have

T^{*} (s, ξ_{1}) \in V_{i} .

If

V_{1}

is an unstable gap, then there are two cases. If for all

1 \leq i \leq n,

we have

V_{i} = V_{1},

then there is nothing to prove. If this is not the case, let

N

be the first positive integer so that

V_{N} \neq V_{1} .

If

V_{1} ⊊ V_{N},

then

V_{N}

is a stable gap by Lemma 5.3, and we again have

V_{i} \subset V_{i + 1}

for all

i \geq N - 1 .

If

V_{N} ⊊ V_{1},

then

V_{N} = T^{*} (s, ξ_{1}) = \partial_{l} V_{1}

by definition and Lemma 5.3. We use Lemma 5.3 again to see that

{V_{i}}_{i \geq N}

is a monotone increasing sequence of closed intervals. In any case, we have

T^{*} (s, ξ_{1}) \in V_{i}

for all

i .

The case when

μ < λ_{1}

can be proved using a similar argument. Thus the lemma is proved for

μ

comparable to

λ_{1} .

Now suppose

μ

is not comparable to

λ_{1} .

We again consider the shortest path from

λ_{1}

to

μ

in

Λ^{*}

similar to above and track how

V_{i}

changes along the path. To illustrate the idea, we consider the following example. Suppose the shortest path from

λ

to

μ

in

Λ^{*}

is of length five:

λ_{1} \to λ_{2} \to \dots \to λ_{5} = μ,

where

λ_{1}, λ_{3}, λ_{5}

are type-1 leaves,

λ_{2}, λ_{4}

are type-0 leaves, and they satisfy

λ_{1} ≲ λ_{2} ≲ λ_{3}

and

λ_{5} ≲ λ_{4} ≲ λ_{3} .

Let

V_{i} = V_{s (λ_{i})} .

We made a turn at

λ_{3}

from the positive flow direction to the negative direction. Our previous discussion shows that

T^{*} (s, ξ_{1}) \in V_{i}

for

i = 1, 2, 3 .

Since

λ_{2}

and

λ_{4}

are incomparable, the core of

I_{λ_{2}}

is contained in a single unstable gap

G

of

I_{λ_{4}} .

The interval

V_{4}

must be the gap

G

containing

V_{3}

by Lemmas 4.14 and 4.15 and the definition of

T .

In particular, we have

T^{*} (s, ξ_{1}) \in V_{4} .

Since

λ_{4}

is negatively adjacent to

λ_{3}

and

V_{3} ⊊ V_{4},

a similar argument to the proof of Lemma 5.3 shows that

V_{4}

is an unstable gap, and an unstable gap will only become larger as we track

V_{i}

backward. Therefore, we have

T^{*} (s, ξ_{1}) \in V_{4} \subset V_{5} .

In general, the path from

λ

to

μ

has a finite number of turns. If we track the interval

V_{i}

along the path, at a turn from the positive direction to the negative direction,

V_{i}

will become a larger unstable gap and can only grow even larger until the next turn happens. Similarly, if we turn from the negative direction to the positive direction,

V_{i}

will become a larger stable gap and can only grow even larger until the next turn happens. Hence,

V_{i}

will be non-decreasing after we make the first turn. But we have shown that

T^{*} (s, ξ_{1})

is in

V_{i}

before we made any turn in the first part of the proof. So the proof of Lemma 5.2 is completed.

Corollary 5.4.

Let $(s, ξ_{1}) \in {LS}^{*}$ be a pointed leftmost section. Then we have $T^{*} (s, ξ_{1}) = ⋂_{λ \in Λ} V_{s (λ)} .$

Proof.

Lemma 5.2 already shows that

T^{*} (s, ξ_{1}) \in ⋂_{λ \in \tilde{ℱ}} V_{s (λ)},

so it suffices to prove that there is some

λ

with

V_{s (λ)}

being a singleton. Suppose

ξ_{1}

is at the infinity of the leaf

λ_{1} .

If

V_{ξ_{1}}

is a single point, it is trivial. If

V_{ξ_{1}}

is a non-trivial closed interval, let

e

be the side of

p (λ_{1})

facing

V_{ξ_{1}},

and let

x_{e}

be the periodic point in

e

as in Proposition 3.1. The periodic orbit

p^{- 1} (x_{e})

intersects some leaf

λ

comparable to

λ_{1} .

If we take a path in

Λ^{*}

from

λ_{1}

to

λ

and record the intervals

V_{i}

as in the proof of Lemma 5.2, there must be some

i

so that

V_{i + 1} ⊊ V_{i} .

Otherwise, we have

V_{1} \subset V_{2} \subset \dots

and so

I_{λ}

has a gap containing

V_{ξ_{1}},

but that contradicts

x_{e} \in p (λ) .

By Lemma 5.3,

V_{i + 1}

is a single point, so is the intersection

⋂_{λ \in \tilde{ℱ}} V_{s (λ)} .

The lemma is proved.

Corollary 5.5.

There is a map $T : LS \to \partial 𝒪$ so that the following diagram commutes:

Proof.

For any

s \in LS,

pick a starting point

ξ

for

s

and define

T (s) = T^{*} (s, ξ) .

By Corollary 5.4, we have

T^{*} (s, ξ) = T^{*} (s, ξ^{'})

for any

(s, ξ), (s, ξ^{'}) \in {LS}^{*} .

Therefore, the map

T

is well defined.

Corollary 5.6.

The map $T$ preserves the cyclic order of elements in $LS .$ In particular, $T$ is injective.

Proof.

The cyclic order of leftmost sections is determined by the cyclic order of their values on embedded lines in the leaf space ([7

D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363

, Lemma 6.25]; see also Section 4). Their images under

T

must follow the same cyclic order by Lemma 5.2.

Lemma 5.7.

The image of $LS$ under $T$ is dense.

Proof.

It is clear that

T

is

π_{1} (M) -

equivariant, so

T (LS)

is a

π_{1} (M) -

invariant subset of

\partial 𝒪 .

The lemma follows from the minimality of the

π_{1} (M) -

action on

\partial 𝒪

[20

S. R. Fenley, Geometry of foliations and flows I: almost transverse pseudo-Anosov flows and asymptotic behavior of foliations. J. Differential Geom. 81 (2009), no. 1, 1–89 Zbl 1160.57026 MR 2477891

].

Lemma 5.8.

The map $T : LS \to \partial 𝒪$ extends continuously to a homeomorphism $T : 𝔖_{left} \to \partial 𝒪 .$

Proof.

By Corollary 5.6, Lemma 5.7 and the fact that

𝔖_{left}

is the completion of

𝒬,

there is a unique continuous extension of

T

to

𝔖_{left},

and the extended

T

is a homeomorphism between

𝔖_{left}

and

\partial 𝒪 .

Proof of Theorem 5.1.

It suffices to show the “moreover” part about the map

T

defined above. The

π_{1} (M) -

equivariance is automatic from the way we define

T .

The structure maps are intertwined by

T

because of Lemma 5.2.

It is also possible to define the universal circle from rightmost sections

𝔖_{right}

by considering the completion of rightmost sections (i.e., the sections of

E_{\infty}

that go rightmost up and leftmost down). In general, there is no reason to expect

𝔖_{left} = 𝔖_{right} .

However, we have the following corollary of Theorem 1.2.

Corollary 5.9.

Under the assumption of Theorem 1.2, the universal circles $𝔖_{left}$ and $𝔖_{right}$ are isomorphic.

Proof.

Using the same proof of Theorem 1.2, it can be shown that

𝔖_{right}

is isomorphic to

\partial 𝒪,

hence isomorphic to

𝔖_{left} .

Corollary 5.9 suggests that one can view the sets of leftmost and rightmost sections as different dense subsets of the same circle. Indeed, the homeomorphisms from

𝔖_{left}

and

𝔖_{right}

to

\partial 𝒪

give embeddings of the sets of leftmost sections and rightmost sections into the

\partial 𝒪,

both with dense image. Corollary 5.4 is true for both embeddings, so leftmost sections and rightmost sections never cross.

6. Invariant laminations

We conclude with a discussion of the invariant laminations on

\partial 𝒪 ≅ 𝔖_{left} .

See Section 2 for a discussion about laminations on a circle.

Any

λ \in Λ

separates

Λ

into two components, the one

Λ^{+} (λ)

containing the flow positive side of

λ

and the one

Λ^{-} (λ)

containing the flow negative side of

λ .

The leaf

λ

also separates

\tilde{M}

into two parts, denoted by

{\tilde{M}}^{+} (λ)

and

{\tilde{M}}^{-} (λ)

with the same sign convention. Define a subset

Ξ^{\pm} (λ)

of

{Symm}_{2} (\partial 𝒪)

by

Ξ^{\pm} (λ) = \partial CH (⋃_{μ \in Λ^{\pm} (λ)} core (I_{μ})) .

The set

Ξ^{\pm}

is then defined as

Ξ^{\pm} = \bar{⋃_{λ \in Λ} Ξ^{\pm} (λ)} .

It is proved in [5

D. Calegari, Promoting essential laminations. Invent. Math. 166 (2006), no. 3, 583–643 Zbl 1106.57014 MR 2257392

] that

Ξ^{\pm}

is indeed a pair of

π_{1} (M) -

invariant laminations on

\partial 𝒪 .

We are now ready to prove Theorem 1.6. Recall that

ℒ_{𝒪}^{s / u}

is the lamination on

\partial 𝒪

induced by

ℱ_{𝒪}^{s / u} .

Proof of Theorem 1.6.

Take any type-0 leaf

λ

and consider the shadow

p (λ) .

Since

λ

is not a fiber, [19

S. R. Fenley, Surfaces transverse to pseudo-Anosov flows and virtual fibers in

3 -

manifolds. Topology 38 (1999), no. 4, 823–859 Zbl 0926.57009 MR 1679801

, Proposition 4.6] shows that

\partial p (λ)

has leaves in both

ℱ_{𝒪}^{u}

and

ℱ_{𝒪}^{s} .

Suppose that

e

is a side of

p (λ)

that is contained in a leaf of

ℱ_{𝒪}^{s}

and consider the leftmost section

s

starting from

Q_{λ} (e) .

If

μ

is a leaf in

Λ^{+} (λ),

we take a path in

Λ^{*}

from

λ

to

μ

and track the closed interval

V_{i}

as in the proof of Lemma 5.2. The interval

V_{0}

is the stable gap of

I_{λ}

facing

e,

and the proof of Lemma 5.2 shows that

V_{i}

is monotone increasing along the path. This means the shadow of

μ

is on the same side of

e

as

p (λ) .

The side

e,

viewed as an element of

{Symm}_{2} (\partial 𝒪),

is therefore in

Ξ^{+} (λ),

hence in

Ξ^{+} .

By transitivity of

ϕ,

every leaf of

ℱ^{s}

is dense in

M .

This implies that the

π_{1} (M) -

image of

\partial e

is dense in

ℒ_{𝒪}^{s}

because

ϕ

has no perfect fits. Since both

Ξ^{+}

and

ℱ_{𝒪}^{s}

are

π_{1} (M) -

invariant and closed, we have

Ξ^{+} \supset ℒ_{𝒪}^{s} .

If

Ξ^{+} - ℒ_{𝒪}^{s}

is not empty, the difference must be a union of diagonals of complementary regions of

ℒ_{𝒪}^{s} .

Note that these diagonals cannot be approximated by leaves in

ℒ_{𝒪}^{s} \cup ℒ_{𝒪}^{u},

so there must be such a diagonal

d

in

Ξ^{+} (λ)

for some

λ .

The corresponding complementary polygon comes from a singular leaf

l

of

ℱ_{𝒪}^{s},

and we denote the singularity in

l

by

s .

Suppose

d

has endpoints

ξ

and

ξ^{'} .

Then there is a sequence of leaves

λ_{n} \in Λ^{+} (λ),

sides

e_{n}

of

p (λ_{n})

and endpoints

ξ_{n}

of

e_{n}

so that

ξ_{n}

converges to

ξ .

Up to taking a subsequence, we can assume that all

e_{n}

are contained in leaves of

ℱ_{𝒪}^{s}

or in leaves of

ℱ_{𝒪}^{u} .

If all

e_{n}

are contained in leaves of

ℱ_{𝒪}^{u},

then

d

cannot be a boundary component of the convex hull of

⋃_{μ \in Λ^{\pm} (λ)} core (I_{μ}),

because

e_{n}

will eventually cross

d

by the absence of perfect fits. So all

e_{n}

are contained in leaves of

ℱ_{𝒪}^{s} .

In particular, the singularity

s

can be approximated by points in shadows of leaves in

Λ^{+} (λ) .

We will show that this is impossible, a contradiction.

If

s

is in

p (λ),

then there are points of

core (I_{λ})

on both sides of

d,

contradicting the assumption that

d \in Ξ^{+} (λ) .

Therefore, the

\tilde{ϕ} -

orbit

p^{- 1} (s)

is disjoint from

λ .

If

p^{- 1} (s)

is contained in

{\tilde{M}}^{+} (λ),

it contradicts our assumption that

d \in Ξ^{+} (λ)

by a similar reason. So

p^{- 1} (s)

is contained in

{\tilde{M}}^{-} (λ) .

Note that

s

is not in

\partial p (λ),

otherwise a face of

l

will be a side of

p (λ),

and the face will be a leaf of

Ξ^{+} (λ)

as we showed above. This again contradicts the assumption that

d \in Ξ^{+} (λ) .

So orbits close enough to

p^{- 1} (s)

will stay in

{\tilde{M}}^{-} (λ),

giving the desired contradiction.

Therefore,

Ξ^{+}

must be the same as

ℒ_{𝒪}^{s} .

For the same reason, we have

Ξ^{-} = ℱ_{𝒪}^{u},

finishing the proof of Theorem 1.6.

Acknowledgements

The author is grateful to his advisor, Yair Minsky, for being inspiring and supportive throughout this project. The author would like to thank Hyungryul Baik, Ellis Buckminster, Danny Calegari, Sergio Fenley, Michael Landry, Anna Parlak and Sam Taylor for helpful comments and conversations. The author thanks the referee for carefully reading the paper and for giving numerous insightful comments and suggestions.

Funding

The author is partially supported by NSF grant DMS-2005328.

References

[1] I. Agol and C. C. Tsang, Dynamics of veering triangulations: infinitesimal components of their flow graphs and applications. Algebr. Geom. Topol. 24 (2024), no. 6, 3401–3453 Zbl 1552.57031 MR 4812222
[2] T. Barbot, Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles. Ergodic Theory Dynam. Systems 15 (1995), no. 2, 247–270 Zbl 0826.58025 MR 1332403
[3] T. Barthelmé, S. Frankel, and K. Mann, Orbit equivalences of pseudo-Anosov flows. Invent. Math. 240 (2025), no. 3, 1119–1192 Zbl 08038253 MR 4902161
[4] T. Barthelmé and K. Mann, Pseudo-Anosov flows: a plane approach. [v1] 2025 [v2] 2026, arXiv:2509.15375v2
[5] D. Calegari, Promoting essential laminations. Invent. Math. 166 (2006), no. 3, 583–643 Zbl 1106.57014 MR 2257392
[6] D. Calegari, Foliations and the geometry of 3-manifolds. Oxford Math. Monogr., Oxford University Press, Oxford, 2007, 363 pp. Zbl 1118.57002 MR 2327361
[7] D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363
[8] A. Candel, Uniformization of surface laminations. Ann. Sci. Éc. Norm. Super. (4) 26 (1993), no. 4, 489–516 Zbl 0785.57009 MR 1235439
[9] J. Cantwell and L. Conlon, Poincaré–Bendixson theory for leaves of codimension one. Trans. Amer. Math. Soc. 265 (1981), no. 1, 181–209 Zbl 0484.57015 MR 0607116
[10] J. Cantwell and L. Conlon, Hyperbolic geometry and homotopic homeomorphisms of surfaces. Geom. Dedicata 177 (2015), no. 1, 27–42 Zbl 1359.37097 MR 3370020
[11] J. Cantwell, L. Conlon, and S. R. Fenley, Endperiodic automorphisms of surfaces and foliations. Ergodic Theory Dynam. Systems 41 (2021), no. 1, 66–212 Zbl 1464.57022 MR 4190052
[12] D. Cooper, D. D. Long, and A. W. Reid, Bundles and finite foliations. Invent. Math. 118 (1994), no. 1, 255–283 Zbl 0858.57015 MR 1292113
[13] S. Fenley, Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry. Geom. Topol. 16 (2012), no. 1, 1–110 Zbl 1279.37026 MR 2872578
[14] S. Fenley and L. Mosher, Quasigeodesic flows in hyperbolic 3-manifolds. Topology 40 (2001), no. 3, 503–537 Zbl 0990.53040 MR 1838993
[15] S. R. Fenley, Asymptotic properties of depth one foliations in hyperbolic $3 -$ manifolds. J. Differential Geom. 36 (1992), no. 2, 269–313 Zbl 0766.53018 MR 1180384
[16] S. R. Fenley, Anosov flows in $3 -$ manifolds. Ann. of Math. (2) 139 (1994), no. 1, 79–115 Zbl 0796.58039 MR 1259365
[17] S. R. Fenley, The structure of branching in Anosov flows of $3 -$ manifolds. Comment. Math. Helv. 73 (1998), no. 2, 259–297 Zbl 0999.37008 MR 1611703
[18] S. R. Fenley, Foliations with good geometry. J. Amer. Math. Soc. 12 (1999), no. 3, 619–676 Zbl 0930.53024 MR 1674739
[19] S. R. Fenley, Surfaces transverse to pseudo-Anosov flows and virtual fibers in $3 -$ manifolds. Topology 38 (1999), no. 4, 823–859 Zbl 0926.57009 MR 1679801
[20] S. R. Fenley, Geometry of foliations and flows I: almost transverse pseudo-Anosov flows and asymptotic behavior of foliations. J. Differential Geom. 81 (2009), no. 1, 1–89 Zbl 1160.57026 MR 2477891
[21] E. Field, H. Kim, C. Leininger, and M. Loving, End-periodic homeomorphisms and volumes of mapping tori. J. Topol. 16 (2023), no. 1, 57–105 Zbl 1567.57023 MR 4532490
[22] D. Fried, Fibrations over $S^{1}$ with pseudo-Anosov monodromy. In Travaux de Thurston sur les surfaces, pp. 51–266, Astérisque 66–67, Société Mathématique de France, Paris, 1979 Zbl 0446.57023 MR 0568308
[23] M. P. Landry, Y. N. Minsky, and S. J. Taylor, Endperiodic maps via pseudo-Anosov flows. 2023, arXiv:2304.10620v1, to appear in Geom. Topol.
[24] M. P. Landry, Y. N. Minsky, and S. J. Taylor, Simultaneous universal circles. J. Topol. 19 (2026), no. 1, article no. e70054, 28 pp. Zbl 08143171 MR 5009301
[25] M. P. Landry and C. C. Tsang, Endperiodic maps, splitting sequences, and branched surfaces. Geom. Topol. 29 (2025), no. 9, 4531–4663 Zbl 08152702 MR 5017754
[26] M. P. Landry and C. C. Tsang, Pseudo-Anosov flows from sutured hierarchies. In preparation
[27] L. Mosher, Dynamical systems and the homology norm of a $3 -$ manifold, I: efficient intersection of surfaces and flows. Duke Math. J. 65 (1992), no. 3, 449–500 Zbl 0754.58030 MR 1154179
[28] L. Mosher, Laminations and flows transverse to finite depth foliations. Preprint, 1996
[29] S. P. Novikov, The topology of foliations (in Russian). Tr. Moskov. Mat. Obs. 14 (1965), 248–278. English translation. Trans. Moscow Math. Soc. 14 (1965), 268–304 Zbl 0247.57006 MR 0200938
[30] A. Parlak, Mutations and faces of the Thurston norm ball dynamically represented by multiple distinct flows. Geom. Topol. 29 (2025), no. 4, 2105–2173 Zbl 08084075 MR 4929474
[31] W. P. Thurston, A norm for the homology of $3 -$ manifolds. Mem. Amer. Math. Soc. 59 (1986), no. 339, 99–130 Zbl 0585.57006 MR 0823443
[32] W. P. Thurston, Three-manifolds, foliations and circles, II. Unfinished manuscript, 1998

Cite this article

Junzhi Huang, Depth-one foliations, pseudo-Anosov flows and universal circles. Groups Geom. Dyn. (2026), published online first

DOI 10.4171/GGD/964

Abstract

1. Introduction

Theorem 1.1.

Theorem 1.2.

Corollary 1.3.

Proof.

Remark 1.4.

Remark 1.5.

Theorem 1.6.

2. Preliminaries

Convention 2.1.

2.1. Pseudo-Anosov flows

Lemma 2.2.

Lemma 2.3.

Convention 2.4.

2.2. End-periodic automorphisms

2.3. Depth-one foliation

Convention 2.5.

Convention 2.6.

2.4. Laminations on S1

3. Infinity of shadows

Proposition 3.1

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Lemma 3.4.

Proof.

Lemma 3.5.

Proof.

Claim.

Proof of the claim.

Lemma 3.6.

Proof.

4. Markers and universal circles

Definition 4.1.

Proof of Theorem 1.1.

4.1. Calegari–Dunfield’s construction

Definition 4.2.

Lemma 4.3 ([7D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363]).

Theorem 4.4.

Theorem 4.5 ([7D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363]).

4.2. Markers contained in a product region

Lemma 4.6.

Proof.

Corollary 4.7.

Proof.

4.3. Markers intersecting a type-0 leaf

Lemma 4.8.

Proof.

Corollary 4.9.

Proof.

Lemma 4.10.

Proof.

Lemma 4.11.

Proof.

Lemma 4.12.

Proof.

Lemma 4.13.

Proof.

Lemma 4.14.

Proof.

Lemma 4.15.

Remark 4.16.

Proof.

5. Building the homeomorphism

Theorem 5.1.

Lemma 5.2.

Proof.

Lemma 5.3.

Proof.

Corollary 5.4.

Proof.

Corollary 5.5.

Proof.

Corollary 5.6.

Proof.

Lemma 5.7.

Proof.

Lemma 5.8.

2.4. Laminations on $S^{1}$

Lemma 4.3 ([7
D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363
]).

Theorem 4.5 ([7
D. Calegari and N. M. Dunfield, Laminations and groups of homeomorphisms of the circle. Invent. Math. 152 (2003), no. 1, 149–204 Zbl 1025.57018 MR 1965363
]).